Moral Hazard, Adverse Selection, and Mortgage Markets

Moral Hazard, Adverse Selection, and Mortgage Markets

by

Barney Paul Hartman-Glaser

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in

Business Administration

in the

Graduate Division

of the

University of California, Berkeley

Committee in charge:Professor Nancy Wallace, Co-chairProfessor Alexei Tchistyi, Co-chair

Professor Dmirty LivdanProfessor Robert Anderson

Spring 2011


Copyright 2011by

Barney Paul Hartman-Glaser

1

Abstract


by

Barney Paul Hartman-GlaserDoctor of Philosophy in Business Administration

University of California, Berkeley

Professor Nancy Wallace, Co-chairProfessor Alexei Tchistyi, Co-chair

This dissertation considers problems of adverse selection and moral hazard in secondarymortgage markets. Chapters 2 and 3 consider moral hazard and adverse selection respec-tively. While Chapter 4 investigates the predictions of the model presented in Chapter 2using data from the commercial mortgage backed securities market.

Chapter 2 derives the optimal design of mortgage backed securities (MBS) in a dynamicsetting with moral hazard. A mortgage underwriter with limited liability can engage incostly effort to screen for low risk borrowers and can sell loans to a secondary market.Secondary market investors cannot observe the effort of the mortgage underwriter, but theycan make their payments to the underwriter conditional on the mortgage defaults. Theoptimal contract between the underwriter and the investors involves a single payment tothe underwriter after a waiting period. Unlike static models that focus on underwriterretention as a means of providing incentives, the model shows that the timing of paymentsto the underwriter is the key incentive mechanism. Moreover, the maturity of the optimalcontract can be short even though the mortgages are long-lived. The model also gives anew reason for mortgage pooling: selling pooled mortgages is more efficient than sellingmortgages individually because pooling allows investors to learn about underwriter effortmore quickly, an information enhancement effect. The model also allows an evaluation ofstandard contracts and shows that the “first loss piece” is a very close approximation to theoptimal contract.

Chapter 3 considers a repeated security issuance game with reputation concerns. Eachperiod, an issuer can choose to securitize an asset and publicly report its quality. However,potential investors cannot directly observe the quality of the asset and a lemons problemensues. The issuer can credibly signal the asset’s quality by retaining a portion of theasset. Incomplete information about issuer type induces reputation concerns which providecredibility to the issuer’s report of asset quality. A mixed strategy equilibrium obtains withthe following 3 properties: (i) the issuer misreports asset quality at least part of the time,(ii) perceived asset quality is a U-shaped function of the issuer’s reputation, and (iii) theissuer retains less of the asset when she has a higher reputation.

2

Chapter 4 documents empirical evidence that subordination levels for commercial mort-gage backed securities (CMBS) depend on issuer reputation in a manner consistent with themodel of Chapter 3. Specifically, issuer retention is negatively correlated with issuer repu-tation. New measures for both issuer reputation and retention are considered.

i

To my father, Barney G. Glaseran inspirational scholar, and

my wife, Tiffany M. Shih,my academic and emotional rock.

ii

Contents

Acknowledgments iv

1 Introduction 1

2 Optimal securitization with moral hazard 32.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Preferences, Technology, and Information . . . . . . . . . . . . . . . . 72.1.2 Optimal Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.4 The benefits of pooling . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Standard contracts and the approximate optimality of the “first loss piece” . 152.2.1 The optimal contract versus a fraction of the mortgage pool . . . . . 152.2.2 The optimal contract versus a “first loss piece” . . . . . . . . . . . . 18

2.3 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.1 An Initial Capital Constraint . . . . . . . . . . . . . . . . . . . . . . 192.3.2 Maturity of the optimal contract . . . . . . . . . . . . . . . . . . . . 212.3.3 Partial Effort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.4 Adverse selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Reputation and signaling in asset sales 283.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.1 Assets, agents and actions . . . . . . . . . . . . . . . . . . . . . . . . 323.2.2 Issuer type, reputation and strategies . . . . . . . . . . . . . . . . . . 333.2.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3.1 The game without reputation . . . . . . . . . . . . . . . . . . . . . . 353.3.2 Reputation dynamics and optimization . . . . . . . . . . . . . . . . . 383.3.3 Separating equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3.4 Mixed strategy equilibria . . . . . . . . . . . . . . . . . . . . . . . . . 423.3.5 Analysis of the mixed strategy equilibrium . . . . . . . . . . . . . . . 48

Contents iii

3.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.4.1 Binary signal space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.4.2 Risky assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.4.3 Path dependent beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4 CMBS issuer reputation and retention 634.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2 Instituational Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.3 Data and Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Bibliography 75

A Proofs for Chapter Two 78

B Proofs for Chapter Three 86

iv

Acknowledgments

First I wish to thank all the faculty who have played a role in producing this dissertation,without their continued support and mentorship this process would have been impossible.In particular, special thanks are due to my co-chairs Nancy Wallace and Alexei Tchistyi. Ithank Nancy for her perpetual willingness to listen to my ideas and help me improve them.I thank Alexei for arriving at the exact right moment and tirelessly helping me develop myskills as a theorist. Dmitry Livdan also deserves thanks on that front, as well as for helpingme prepare for the job market. Other faculty whose support deserves special mention areRobert Anderson, Willie Fuchs, Robert Helsley, Dwight Jaffee, Atif Mian, Christine Parlour,Tomasz Piskorski, Jacob Sagi, and Steve Tedalis. Finally, Darrell Duffie deserves specialthanks for early inspiration and support on the job market; without taking his course indynamic asset pricing theory, I would not have considered a Ph.D. in finance.

Next, I wish to thank those fellow graduate students who greatly influenced my devel-opment as a scholar. My cohort of Javed Ahmed, Bradyn Breon-Drish, Andres Donangelo,and Nishanth Rajan were particularly helpful in navigating the mine field that is gettinga Ph.D. I learned so much from all of them. Andres was a great help as I prepared forthe job market by helping me achieve the right balance of stress and excitement about thewhole process. Sebastian Gryglewicz has influenced my thinking perhaps most of all, andhas constantly been available to discuss ideas.

Finally, my family deserves my grateful acknowledgment. My parents, Barney andCarolyn Glaser, and Gaylord and Susan Fukumoto have always encouraged and supportedme, even when I was a difficult adolescent (between the ages of 14 and 29). My father,Barney, has instilled in me a love of learning and scholarship that will always be central tomy life. This is perhaps the greatest gift a father can bestow upon his son. And last but notleast, my wife Tiffany Shih deserves the most thanks of all. She did more than I thoughtwas possible to help me finish this dissertation, including but not limited to proof readingmultiple drafts, providing constructive conceptual comments, walking the dog when I wasaway/too tired, listening to my fears and self doubts, cooking my favorite meals, and thislist goes on.

1

Chapter 1

Introduction

Financial economists have worried about problems of asymmetric information, such asmoral hazard (MH) and adverse selection (AS), in financial markets for decades. Recentevents, like the subprime default crisis, seem to indicate that these problems are particularlypronounced in mortgage markets. At the same time, mortgage markets have a structure thatposes new challenges for the both the theoretical and empirical literatures dealing with MHand AS. Moreover, mortgage markets are some of the largest financial markets in existenceand are essential to the efficiency of the real economy. Thus, a deeper understanding of thespecific manifestations and implications of MH and AS for mortgage markets is in order.In the following chapters I will consider 2 important examples of how these issues may beintrinsically different from those considered in the past literature. I will also present newempirical evidence from the CMBS market.

In Chapter 2, I consider the problem of providing incentives to a mortgage underwriter,in which effort has long term consequences and must be exerted over many tasks at once.1

A mortgage underwriter with limited liability can engage in costly effort to screen for lowrisk borrowers and can sell loans to a secondary market. Secondary market investors can-not observe the effort of the mortgage underwriter, but they can make their payments tothe underwriter conditional on the mortgage defaults. The optimal contract between theunderwriter and the investors involves a single payment to the underwriter after a waitingperiod. Unlike static models that focus on underwriter retention as a means of providingincentives, the model shows that the timing of payments to the underwriter is the key incen-tive mechanism. Moreover, the maturity of the optimal contract can be short even thoughthe mortgages are long-lived. The model also gives a new reason for mortgage pooling: sell-ing pooled mortgages is more efficient than selling mortgages individually because poolingallows investors to learn about underwriter effort more quickly, an information enhancementeffect. The model also allows an evaluation of standard contracts and shows that the “firstloss piece” is a very close approximation to the optimal contract.

1Chapter 2 draws on the co-authored article “Optimal Securitization with Moral Hazard” with AlexeiTchistyi and Tomazs Piskorski. At the time of submission of this dissertation, the article had not beenpublished.

Chapter 1. Introduction 2

In Chapter 3, I consider an adverse selection problem faced by investors when an issuercan both signal her private information through partial retention and build a reputation forhonest reporting. The model can be characterized as a repeated security issuance game withreputation concerns. Each period, an issuer can choose to securitize an asset and publiclyreport its quality. However, potential investors cannot directly observe the quality of the assetand a lemons problem ensues. The issuer can credibly signal the asset’s quality by retaininga portion of the asset. Incomplete information about issuer type induces reputation concernswhich provide credibility to the issuer’s report of asset quality. A mixed strategy equilibriumobtains with the following 3 properties: (i) the issuer misreports asset quality at least partof the time, (ii) perceived asset quality is a U-shaped function of the issuer’s reputation, and(iii) the issuer retains less of the asset when she has a higher reputation.

Chapter 4 documents empirical evidence that subordination levels for commercial mort-gage backed securities (CMBS) depend on issuer reputation in a manner consistent with themodel of Chapter 3. Specifically, issuer retention is negatively correlated with issuer repu-tation. New measures for both issuer reputation and retention are considered.

Note that in Chapter 2, I use the pronoun “we” to indicate that the work was done bymultiple people whereas in Chapter 3 I used the pronoun “I,” as that work is entirely myown. Also note that although some symbols are present in both Chapters 2 and 3, theirmeaning may differ across chapters.

3

Chapter 2

Optimal securitization with moralhazard

Mortgage underwriters face a dilemma: either to implement high underwriting stan-dards and underwrite only high quality mortgages or relax underwriting standards in orderto save on expenses. For example, an underwriter can collect as much information as possibleabout each mortgage applicant and fund only the most credit-worthy borrowers. Alterna-tively, an underwriter could collect no information at all and simply make loans to everymortgage applicant. Clearly, the second approach, while less costly in terms of underwritingexpenses, will result in higher default risks for the underwritten mortgages. Moreover, mort-gage underwriters typically wish to sell their loans in a secondary market rather than holdloans in their portfolio. Investors do not observe the underwriter’s effort and consequentlydo not observe the quality of the mortgages they are buying.1

The recent mortgage crisis has brought a lot of attention to the potential agency con-flict arising from the separation of loan’s originator and the bearer of the loan’s default risk.Policy-makers and market observers have emphasized that the originators of loans and theunderwriters of mortgage-backed securities might have lacked proper incentives to act in thebest interests of investors, and as a consequence this possible misalignment of incentivesmight have importantly contributed to the mortgage default crisis. Ultimately, these con-cerns resulted in a number of policy proposals, beginning on June 2009 when the ObamaAdministration released its “Financial Regulatory Reform” proposal which calls for loanoriginators or sponsors to retain a part of the credit risk of securitized assets.2 However,beside a general notion that aligning incentives requires the participants in securitization

1Recent empirical evidence in Keys, Mukherjee, Seru, and Vig (2008) suggests that securitization mighthave adversely affected the screening incentives of lenders. For an good description of the market forsecuritized subprime loans, see Ashcraft and Schuermann (2009).

2More precisely, the proposal requires originators or sponsors to retain a material portion (generally 5%)of the credit risk of securitized assets and prohibits “hedging or otherwise transferring” the retained risk. See“Financial Regulatory Reform, A New Foundation: Rebuilding Financial Regulation and Supervision,” De-partment of the Treasury, pages 44-45, at http://www.financialstability.gov/docs/regs/FinalReport web.pdf

Chapter 2. Optimal securitization with moral hazard 4

process to hold an economic interest in the credit risk of securitized assets (or “skin in thegame”), little is known about how to design these provisions efficiently, especially fully rec-ognizing the long term duration of assets in question. Consequently, this question has beenthe subject of significant and ongoing discussions among the Administration, Congress, reg-ulators and market participants.3 Our paper aims to inform this debate by examining howthe design of mortgage backed securities (MBS) can efficiently address this agency conflict.

We consider an optimal contracting problem between a mortgage underwriter andsecondary-market investors. At the origination date, the underwriter can choose to un-dertake costly effort that results in low expected default rates for the underwritten mort-gages. If the underwriter chooses to shirk, the mortgages will have a high expected defaultrate. Thus, the effort technology of our model results in mortgage performance that occursthrough time, rather than on a single date as in the previous literature. In addition to costlyhidden effort, we include a motivation to securitize by assuming that by selling mortgages,rather than holding them in her portfolio, the mortgage underwriter can exploit new invest-ment opportunities, i.e., underwrite more mortgages. We model this feature by assumingthat the underwriter is impatient, as in DeMarzo and Duffie (1999), so that the underwriterhas a higher discount rate than the investors.4 Investors do not observe the actions of theunderwriter, however the timing of mortgage defaults is publicly observable and contractible.

We derive the optimal contract between the underwriter and the investors that imple-ments costly effort and maximizes the expected payoff for the underwriter, provided theinvestors are making non-negative profits in expectation. We do not make restrictive as-sumptions on the form of the contract. Instead, we include all possible payment schedulesbetween the investors and the underwriter in the space of admissible contracts, so long asthey depend only on the realization of mortgage defaults and provide limited liability to theunderwriter. This setup, which allows information to be revealed over time, allows us toaddress a central issues in the market for MBS. Namely, how does the fact that mortgageperformance occurs through time affect the contracting problem between the investors andthe underwriter? Moreover, how much time is needed before the investors can completelypay off the underwriter while maintaining incentive compatibility?

Despite the apparent complexity of the contracting problem, the optimal contract takesa simple form: The investors receive the entire pool of mortgages at time zero and make asingle lump sum transfer to the underwriter after a waiting period provided no default occurs.If a single default occurs during the waiting period, the investors keep the mortgages, butno payments are made to the underwriter.

The timing and structure of the optimal contract arise from a trade-off between twoforces. Delaying payment to the underwriter results in a more precise signal on the under-writer’s action, which lowers the cost of incentive provision. On the other hand, delayingpayment is costly due to the relative impatience of the underwriter. By making the payment

3See American Securitization Forum (2009).4An alternative motivation for the relative impatience of underwriters are regulatory capital requirements

which induce a preference for cash over risky assets.


for mortgages contingent upon an initial period of no default, the investors can provide in-centives for underwriters to underwrite low risk mortgages since high risk mortgages will bemore likely to default during the initial waiting period. However, delaying payments pastthe initial waiting period is suboptimal since the underwriter is impatient.

Interestingly, the optimal contract calls for the underwriter to pool mortgages ratherthan sell each mortgage individually. By observing the timing of a single default, the investorslearn about the quality of the remaining mortgages. As a result, the investors can infer thequality of the mortgages sooner by observing the entire pool rather than a single mortgageat a time, which we call the information enhancement effect. By making payment contingentupon the performance of the entire pool rather than each individual mortgage, it is possibleto speed up payment to the underwriter while maintaining incentive compatibility. Thisresult is not driven by any benefits from diversification.

Our findings are in contrast with the previous literature on security design with asym-metric information that primarily focuses on a static setting, e.g., DeMarzo (2005). Weshow that the timing of payments plays an important role when the information about theunderlying assets is revealed over time. In the dynamic setting of this paper, the optimalcontract is about when the underwriter is paid, rather than what piece of the underlyingassets it retains.

Our paper relates most closely to the literature on optimal security design and assetbacked securities (ABS). One approach of this literature is to treat the security design prob-lem faced by issuers as a standard capital structure problem as characterized by Myers andMajluf (1984) giving rise to a pecking order theory of asset backed securities. Nachmanand Noe (1994) present a rigorous frame work for when a given a security design minimizesmispricing due to asymmetric information, showing standard debt is optimal over a verybroad class of security design problems. Building on the pecking order intuition, Riddiough(1997a) shows that an informed issuer can increase her proceeds from securitization by cre-ating multiple securities, or tranches, with differing levels of exposure to the issuer’s privateinformation. Moreover, pooling assets that are not perfectly correlated can provide somediversification benefits and thus reduce the lemons discount.

Another approach to the optimal design of asset backed securities considers the role ofcostly signaling. The basic intuition of this approach is that an issuer of ABS can signalher private information by retaining a fraction of the issued security as in Leland and Pyle(1977). Building on this intuition, DeMarzo and Duffie (1999) presents a model of securitydesign where an issuer minimizes the cost of signaling her private information by choosinga security design. Applying the security design framework of DeMarzo and Duffie (1999),DeMarzo (2005) explains the pooling and tranching structure of ABS. In his model, an issuerof ABS can signal her private information about a pool of assets by retaining a fraction ofa security which is highly sensitive to that information. This signaling mechanism explainsthe multi-class, or tranched structure, of ABS.

Other studies of asset backed securities have focused on different types of asymmetricinformation. For example, Axelson (2007) considers a setting in which investors have su-perior information about the distribution of asset cash flows. The author gives conditions


under which pooling may be an optimal response to investor private information and forwhich single asset sale is preferred.

The literature on security design and ABS presented above utilizes mostly one periodmodels of securitization. In contrast to the previous literature, we take a different approachby modeling mortgages which can default after some time has elapsed. This additional aspectallows us to show that the timing of payments from mortgage securitization can be a keyincentive mechanism and that the duration of the optimal contract can be short while theduration of the mortgages is long.

Another important difference between our model and the literature is that there is littleprevious work on costly hidden actions in underwriting practices. An exception is Innes(1990), who considers a one period moral hazard model of security design. Other work thatconsiders a security design in the context of a moral hazard framework include DeMarzo andSannikov (2006) and DeMarzo and Fishman (2007), which both consider long term contractsfor repeated agency conflicts in which actions have short term consequences. In our setting,the agent may only take actions at a single point in time, however those actions have longterm consequences.

The moral hazard problem in underwriting practices is likely to be important in privatesecuritization markets where both the quality of assets and the operations of issuers areextremely difficult to verify. Indeed, some empirical studies, such as Mian and Sufi (2009),suggest that “mis-priced agency conflicts” may have played a crucial role in the currentmortgage crisis. In addition, evidence presented in Keys, Mukherjee, Seru, and Vig (2008)suggests that securitization of subrpime loans led to lax lending standards, especially whenthere is “soft” information about borrowers which determines default risk but is not easilyverifiable by investors.

Our paper also closely relates to the literature on “multi-task” Principal-Agent prob-lems, for example Bond and Gomes (2009) and Laux (2001), and to the literature onPrincipal-Agent problems where actions have persistent effects, for example Abreu, Mil-grom, and Pearce (1991) and Sannikov and Skrzypacz (2010). The problem of providingincentives to a mortgage underwriter can be viewed as a multitask contracting problemsince the underwriter must evaluate multiple loans. Bond and Gomes (2009) analyzes a verybroad class of “multi-task” Principal-Agent problems and finds that optimal contracts formulti-task problems take the form of cutoff rules analogous to the optimal contract in oursetting. Our results differ from this literature in that we emphasize the delayed nature ofinformation revelation inherent in the specific problem of providing incentives to mortgageunderwriters. It is this feature which provides a contribution to the literature on agencyproblems in which actions have persistent effects, as in Abreu, Milgrom, and Pearce (1991).In this literature, increasing the lag between when an action is taken and output is observed,can increase efficiency when discount rates are small. This effect arises due to a statisticalinference effect that is similar to our information enhancement effect. In our setting, the lagbetween actions and outcomes is stochastic and has a distribution that depends on hiddenactions. Therefore, the key parameters which drive efficiency are the difference betweenexpected information lags and the difference in discount rates. Moreover, contract efficiency


increases when either difference decreases. Although we specifically frame the contractingproblem in terms of a mortgage underwriter and investors in mortgage backed securities,the model could apply to other problems in which an agent takes multiple hidden actionsthat have persistent consequences, for example a CEO deciding the allocation of capital tomultiple long lived projects within a single firm.

2.1 The Model

2.1.1 Preferences, Technology, and Information

Time is infinite, continuous, and indexed by t. A risk-neutral agent (the underwriter5)originates N mortgages that she wants to sell to a risk-neutral principal (the investors)immediately after origination. The underwriter has the constant discount rate of γ and theinvestors have the constant discount rate of r. We assume γ > r. This assumption couldproxy for a preference for cash or additional investment opportunities of the underwriterDeMarzo and Duffie (1999).

The underwriter may undertake an action e ∈ 0, 1 at cost C ·e at the origination of thepool of mortgages (t =0). This action is hidden from the investors and hence not contractible.Each mortgage generates constant coupon u until it defaults. Individual mortgages defaultaccording to an exponential random variable with parameter λ ∈ λH , λL such that λ = λLif e = 1 and λ = λH if e = 0 and λH > λL. Upon default, all assets pay a lump sum recoveryof R < u/r. All defaults are mutually independent conditional on effort. It may seem overlysimplistic to assume that low underwriter effort leads to only high risk mortgages. A morerealistic assumption is that low effort to leads to a mixture of both high risk and low riskmortgages. Such a setup would complicate the analysis as the mixture of two exponentialdistributions is not itself exponential. However, as we argue below, it does not add richnessto the model to assume that low effort leads to a mixture of mortgage risk types.

A contract consists of transfers from the investors to the underwriter depending onmortgage defaults. Specifically, let Dt denote the total number of defaults that have occurredby time t and Ft the filtration generated by Dt. It will also be convenient to define thefollowing sequence of stopping times

τn = inft : Dt ≥ n.

for n ≥ 0. Also let τ0 = 0 for convenience. Formally, a contract is an Ft-measurable processXt giving the cumulative transfer to the underwriter by time t so that dXt denotes theinstantaneous transfer to the underwriter at time t. Readers unfamiliar with this notationcan think of Xt as a function of time t and all previous default times τn ≤ t so that dXt =xn(t, τ0, . . . , τn)dt for some family of functions xn(·)n≤N such that Xt is adapted to Ft. We

5Although we refer to the agent in our model as the underwriter, our setting applies equally well to otheractors than mortgage underwriters. The defining characteristic of the agent in our model is that she canundertake costly hidden action to screen out high risk mortgages or assets.


restrict our attention to contracts that satisfy the limited liability constraint dXt ≥ 0 andare absolutely integrable. The underwriter thus has the following utility for a given contractXt and effort e

E

[∫ ∞0

e−γtdXt

∣∣e]− Ce.All integrals will be Stieljes integrals.6

2.1.2 Optimal Contracts

We assume that implementing high effort is optimal, hence the investors’ problem is tomaximize profits subject to delivering a contract that provides incentives to expend effortand a certain promised level of utility to the underwriter. We call a such contracts incentivecompatible. Once we restrict our attention to incentive compatible contracts, the value theinvestors place on holding the mortgages is fixed since the contract cannot affect the dis-tribution of mortgage defaults other than to guarantee that the underwriter only originateslow risk mortgages. Hence, the investors maximize profits by choosing the incentive com-patible contract with the lowest expected cost under their discount rate. In other words, theinvestor chooses the least costly incentive compatible contract. We state this formally in thefollowing definition.

Definition 1. Given a promised utility a0 to the underwriter, a contract Xt is optimal if itsolves the following problem

b(a0) = mindXt≥0

E

[∫ ∞0

e−rtdXt

∣∣e = 1

](I)

such that

E

[∫ ∞0

e−γtdXt

∣∣e = 0

]≤ E

[∫ ∞0

e−γtdXt

∣∣e = 1

]− C. (IC)

and

a0 ≤ E

[∫ ∞0

e−γtdXt

∣∣e = 1

]− C (PC)

It is important to note that Definition 1 is equivalent to a definition where we hold thecost to the investor fixed and maximize the utility of the underwriter. As an alternativeto Definition 1, we could fix the cost paid by the investor b and find the contract whichmaximizes the underwriters initial utility a0. In the analysis that follows, we will find aone-to-one relationship between the cost paid by the investors and the value delivered to theunderwriter; for any level of initial promised utility of the underwriter, we know the costpaid by the investors under the optimal contract and vice versa.

6Readers unfamiliar with Stieljes integrals may simply view this integral for an arbitrary integrand g as∫g(t)dXt =

∫g(t)x(t)dt where x(t) is the time t change in X(t). For a reference see Carter and Van Brunt

(2000).


2.1.3 Solution

The contracting problem stated thus far amounts to solving an infinite dimensional op-timization problem which at first glance seems quite complicated. In this section we willgive a heuristic argument that transforms, subject to verification, our contracting problemto a simple problem of solving two equations for two unknowns by making a series of in-tuitive guesses about the form of the optimal contract. This argument follows from theobservation that the optimal contract should feature the most front-loaded payment sched-ule which maintains incentive compatibility. It does not, however, constitute a rigorous proofof the main result. The proof requires the slightly more sophisticated, yet still quite simple,observation that a useful characterization of the class of incentive compatible contracts ob-tains by relating the underwriter’s expected value for the contract under high and low effortvia a linear approximation of the Radon-Nikodym derivative of the respective probabilitymeasures.

We start by giving a heuristic derivation of the optimal contract when N = 2. Thisbase case provides the basic intuition we will use throughout our solution. Some paymentmust be contingent on τ1 and τ2 to provide incentives to exert effort. If all payment occursregardless of the realization of τ1 and τ2, then there is no incentive for the underwriter toexert effort. Specifically, the contract should reward the underwriter when τ1 and τ2 arerelatively larger and punish the underwriter when τ1 and τ2 are relatively smaller since τ1

and τ2 are more likely to be large when the underwriter exerts effort. At the same time, theoptimal contract should completely pay off the underwriter as quickly as possible to exploitthe difference in discount rates of the investor and underwriter.

Given the intuition above, it is clear that the optimal contract will balance providingincentives with front loading payment. Observe that we can always take an arbitrary in-centive compatible contract and move some payment that occurs later to an earlier date.To maintain incentive compatibility, we must then move some payment that occurs earlierand move it to a later date. Doing so repeatedly should move all payment to a single datestrictly later than t = 0. It is not yet clear that this process will result in reducing the costof the contract, but we use it as a starting point. To that end, we focus on contracts of theform dXt = 0 for t 6= t0 and dXt0 = I((τ1, τ2) ∈ A)y0 for some t0 ≥ 0 and set of events Achosen such that Xt is Ft-measurable.7 We further suppose that both the participation andincentive compatibility constraints bind.8 We then have the following two equations

e−γt0y0P ((τ1, τ2) ∈ A|e = 1) = a0 + C, (2.1)

e−γt0y0P ((τ1, τ2) ∈ A|e = 0) = a0, (2.2)

where P (·|e) denotes probability conditional on effort. Such a contract will cost the investors

e−rt0y0P ((τ1, τ2) ∈ A|e = 1) = e−(r−γ)t0(a0 + C), (2.3)

7I(·) is the indicator function.8In fact, the participation constraint may be slack at the optimal contract. However for the purpose of

this informal derivation the assumption that the participation constraint binds will turn out to be innocuous.


which is increasing in t0 and independent of A. This implies that the optimal contract ofthis form will choose the set A to minimize t0 subject to satisfying equations (2.1) and (2.2).To do so, we choose A such that I((τ1, τ2) ∈ A) depends only on τ1 since any dependenceon τ2 will require an increase in t0 to maintain incentive compatibility. We will return tothis point after stating the optimal contract. Moreover, we guess that A should take on assimple structure as possible. So let A = τ1 ≥ t0, then equations (2.1) and (2.2) reduce tothe following

e−(γ+2λL)t0y0 = a0 + C, (2.4)

e−(γ+2λH)t0y0 = a0. (2.5)

We can easily solve (2.4) and (2.5) for t0 and y0. We formally state the optimal contract(which provides incentives for effort) in the following proposition.

Proposition 1. Let

a0 = max

a0,

C(γ − r)N(λH − λL)

(2.6)

An optimal contract Xt that implements high effort is given by

dXt =

0 if t 6= t0,y0I(τ1 ≥ t0) if t = t0,

(2.7)

where

t0 =1

N(λH − λL)log

(a0 + C

a0

), (2.8)

y0 =

(a0 + C

a0

) γ+NλLN(λH−λL)

(a0 + C). (2.9)

Moreover

b(a0) = (a0 + C)

(a0 + C

a0

) γ−rN(λH−λL)

. (2.10)

The contract calls for no transfers from the investors to the underwriter to take placeuntil the time t0 given by equation (2.8). If the first default time τ1 ≥ t0 then the the contractcalls for a payment of y0 given by equation (2.9) at time t0. Equation (2.8) is the productof two terms. The first term is the inverse of the difference between the arrival intensity ofτ1 given low effort and the arrival intensity of t0 given high effort. The second term is thedifference in the logs of the present value (gross of the cost of effort) of the contract fromhigh effort and low effort respectively. Thus, t0 is set so that the expected present value ofa transfer of y0 at t0 under the underwriter’s discount rate is exactly equal to a0 + C underhigh effort, and a0 under low effort.

The cost of the contract to the investors is given by b(a0) in equation (2.10) and is theproduct of two terms. The first term is the expected present value of transfers under the


discount rate of the underwriter. The second term adjusts this expected present value tothe discount rate of the investors. Not that when a0 < a0, the optimal contract delivers apayoff to the investor greater than necessary to satisfy the participation constraint. Thisis because for small a0, the time required to make the participation constraint bind, whileusing a contract of the proposed form, would be very long, and thus destroy some socialsurplus.

The intuition behind Proposition 1 lies in the tradeoff between two forces: the cost ofwaiting effect, and the wealth transfer effect. On the one hand, the cost of waiting effect arisesbecause delaying payment is costly, in fact reduces total surplus, due to the difference in thediscount rates of the underwriter and investors. On the other hand, the incentive compatibleconstraint implies a minimum sensitivity of contracted payments to mortgage performancewhich is greater when the contract conditions solely on more noisy, i.e. earlier, signals.At the same time, the limited liability constraint places a lower bound on these payments.This interaction leads to the wealth transfer effect: contracts that condition solely on earlyinformation imply the underwriter must receive a payoff greater than necessary to satisfy herparticipation constraint. The optimal contract then balances the cost of waiting effect withthe wealth transfer effect. This trade-off results in a type of “cutoff” rule similar to that ofthe optimal contract in Bond and Gomes (2009): if the performance of the mortgages passessome threshold then the underwriter is compensated, otherwise the underwriter is punished.

The strategy of the proof of Proposition 1 is to find a weaker condition than (IC) whichis more convenient to work with. We then show the proposed contract is optimal over thespace of contracts satisfying this weaker condition and verify that it satisfies (IC) and (PC).Note that the underwriter’s expected value of the contract under low effort is related to theexpected underwriter value of the contract under high effort via a change of measure. If welet Q be the measure induced by low effort and P be the measure induced by high effort,then an application of the monotone convergence theorem leads to the following equality:

E

[∫ ∞0

e−γtdXt|e = 0

]= E

[∫ ∞0

e−γtπ(t)dXt|e = 1

], (2.11)

where πt = dQdP is the Radon-Nikodym derivative of Q with respect to P.9 Such a change of

measure allows one to write the investors problem entirely in terms of conditional expecta-tions with respect to e = 1. Unfortunately, direct calculation of π(t) is not possible, howeverwe do have the following inequality for an arbitrary incentive compatible contract

E

[∫ ∞0

e−γtdXt|e = 0

]≥ E

[∫ ∞0

e−γte−N(λH−λL)tdXt|e = 1

](2.12)

≥ a0

a0 + CE

[∫ ∞0

(N(λH − λL)(t0 − t) + 1)e−γtdXt|e = 1

]. (2.13)

9This is a slight abuse of notation, but the reader familiar with change of measure can think of πt as therestriction of dQ

dP to Ft the filtration generated by Dt.


ΠHΩ2L

ΠHΩ2L

ΠHΩ3L

e-NHΛH-ΛLL t

t0t

a0+C

a0

Π

Figure 2.1: A plot of realizations of the Radon-Nikodym derivative π(t) for different samplepaths ω1, ω2, and ω3. The thick curve is the lower bound on the Radon-Nikodym for allpossible sample paths. The straight line is a linear approximation to the lower bound. Sincethe lower bound is convex, it dominates the straight line and we can use the straight line tofind an inequality relating the expectations of underwriter value under high and low effort.This inequality is a useful characterization of the set of incentive compatible contracts.

Inequality (2.12) arises from approximating π(t) and is verified in the appendix. Inequality(2.13) arises from the fact that e−N(λH−λL)t is convex in t and hence can be bounded belowby a linear function of t. Figure 2.1 gives graphical intuition of the argument that yields thisinequality.

Inequality (2.13) allows us to approximate the incentive compatibility constraint interms of conditional expectations with respect to e = 1. For the purpose of exposition,assume the participation constraint (PC) binds. This allows us to combine the incentivecompatibility constraint (IC) and participation constraint (PC) (which binds) to get thefollowing useful sufficient condition for an arbitrary contract to be incentive compatible

1

a0 + CE

[∫ ∞0

te−γtdXt|e = 1

]≥ t0. (2.14)

Inequality (2.14) shows that the minimum duration of any incentive compatible contract isexactly t0, which turns out to be the duration of the optimal contract. Hence, the proof of


Proposition 1 shows that optimal contracting problem we consider comes down to a durationminimization problem.

In order to find the minimum-duration contract we note that we can always improve onan arbitrary incentive compatible contract by delaying payment that occurs before t0 andaccelerating payment that occurs after t0 until all payment occurs at t0. Doing so reducesthe duration of the contract. Eventually we will have all payment occurring at time t0. Theresulting contract is the optimal contract stated in Proposition 1.

To understand why the optimal contract only uses the information revealed by timet0 as opposed to waiting to gather more information, it is useful to think of the investors’problem as a standard hypothesis testing problem in which there is a trade off between testpower and the period of observation required to perform the test. At each point in time,the investors essentially test the null hypothesis H0 : the underwriter chose e = 1 versus thealternative hypothesis H1 : the underwriter chose e = 0. They then pay the underwriterbased on the outcome of the test. However, they must choose the tests and payments toprovide incentives to the underwriter while maintaining limited liability in the least costlymanner.

For the purpose of exposition, let us consider the following alternative to the optimalcontract

dXt = y0I(accept H0 given Dss≤t0 , t = t0),

where H0 is accepted if Dss≤t0 ∈ A for some A ⊂ Ft0 . Incentive compatibility implies thatthis contract must correspond to a likelihood ratio test which accepts the null hypothesis if

P (Dss≤t0 ∈ A|e = 1)

P (Dss≤t0 ∈ A|e = 0)=a0 + c

a0

. (2.15)

Equation (2.15) illustrates a bit of classic principal-agent intuition in the optimal contractingproblem we consider. Incentive compatibility imposes a trade-off between the significancelevel and power of the likelihood ratio test used by the investors to determine payments tothe underwriter. This implies that if the investors use a more powerful test than the testemployed in the optimal contract, for instance a test which uses more information than thefirst default time, then the test must reject the null hypothesis at a lower significance levelthan the optimal contract. In other words, if the investors wish to condition payments tothe underwriter on more information than used by the optimal contract, then the contractmust be less strict in order to satisfy incentive compatibility. Accordingly, such a test willhave to use a longer period of observation than the optimal contract, in other words t0 ≥ t0.Since part of the surplus in the model arises from front loading payments to the underwriter,any alternative of the above form will be suboptimal. Hence, by adding a time dimension tothe way information is revealed in the model, we emphasize the trade off between the powerof the test employed in the contract, and the amount of observation time needed to performthe test and preserve incentive compatibility.

In our model, the trade-off between the cost of waiting and the benefit of waiting(better information quality) is driven by the limited liability of the underwriter. One could


implement high effort with contract with duration less than t0, that is by using a test whichtakes less time to implement than the optimal contract. However, using such a test requiresthe test be less powerful. Moreover, the most the agent can be punished for a rejection ofthe null hypothesis is a zero payoff. This in turn implies that a less powerful test must paythe underwriter more in the event that the null is not rejected in order to maintain incentivecompatibility. Thus, such a contract will feature a slack participation constraint and will besuboptimal.

The contract given in Proposition 1 is slightly stark in that it severely punishes theunderwriter if even one mortgage defaults prior to time t0. However, an alternative interpre-tation of Proposition 1 is that it provides an upper bound on the efficiency of an arbitraryincentive compatible contract. In this sense, the result is very useful for evaluating somestandard alternative contracts, an exercise we take up in Section 2.2.

2.1.4 The benefits of pooling

One important feature of MBS is the process of “pooling.” In this process, an issuer ofan MBS bundles together many mortgages to form a collateral pool. This security designcontrasts with individual loan sale in which an issuer simply sells each loan separately.Individual loan sale means that the transfers corresponding to the sale of one loan cannotaffect the transfers corresponding to the sale of another. Hence, in the context of our model,individual loan sales correspond to a contract which is the sum of N individual contracts,each of which depends on only one mortgage. Let Wt denote a contract which calls forindividual mortgage sale, then

dWt =N∑n=1

dXnt ,

where Xnt is the payment made to the underwriter for the sale of an individual mortgage.

Since each mortgage payoff is independent and identically distributed after the underwriterchooses effort, it is natural to only consider individual loan sale contracts, which imply Wt

is measurable with respect to the filtration generated by Dt. Thus, individual loan salecontracts are contained in the contract space we consider in the derivation of the optimalcontract. This fact leads us to state the following important corollary to Proposition 1.

Corollary 1. Pooling mortgages is more efficient than individual loan sale. Specifically,individual loan sale contracts are more costly to provide than the contract of Proposition 1.

Notice that Corollary 1 does not depend on the number of mortgages, N , in the pool.Moreover, the benefits of pooling do not arise from risk diversification benefits. Otherresults in the literature, for example DeMarzo (2005), who considers an informed issuerselling multiple assets, attribute the benefits of pooling to the so called risk diversificationeffect.10 In his model, if some portion of the payoff from assets is unrelated to the private

10In Diamond (1984) a financial intermediary benefits from lending to multiple entrepreneurs due to therisk diversification effect.


information of the issuer, then under mild assumptions on the distribution of this residualrisk, the issuer can create a security with less risky payoffs than the pure pass through pool,in other words a senior “tranche.” The issuer can then signal her private information byretaining a portion of the residual payoffs.

In our model, pooling is a consequence of providing incentives in the least costly manner.Returning to the intuition gained by viewing the contracting problem as a hypothesis testingproblem, pooling in our model trades off the time it takes to implement the test with the lossin power from ignoring all defaults that occur after the first default. We call the decreasedtime required to implement the test the information enhancement effect of pooling. A similarconcept is present, although in a static setting, in Laux (2001), in which a manager’s limitedliability constraint can be relaxed by creating a contract that is contingent on multipleoutcomes. The main difference between those results and our result is the channel by whichpooling outcomes decreases the shadow price of limited liability. In our model the key isreducing the time necessary to implement a given test as opposed to minimizing the totalpunishment necessary to provide incentives.

The information enhancement effect in our setting also bears resemblance to the statis-tical inference effect of Abreu, Milgrom, and Pearce (1991). In that paper, increasing lagsbetween initial actions and eventual outcomes increases the amount of information availableto perform inference.

2.2 Standard contracts and the approximate optimal-

ity of the “first loss piece”

An interesting question to ask is how closely we can approximate the optimal contractusing an alternative, and perhaps more standard, contract. To answer this question, wecompare the optimal contract to two possible alternative contracts, one in which the under-writer retains a pure fraction of the mortgage pool, and one in which the underwriter retainsa fraction of a “first loss piece.”.

2.2.1 The optimal contract versus a fraction of the mortgage pool

First we consider contracts in which the underwriter retains an fraction of the pool ofmortgages and receives a lump sum transfer at time t = 0. Note that the total cash flowfrom the pool of mortgages at time t is u(N −Dt)dt + RdDt. Hence a contract which callsfor the underwriter to receive a time zero cash payment of K and retain a fraction α of thepool of mortgages must take the following form

dXt =

K t = 0α(u(N −Dt)dt+RdDt) t ≥ 0

. (2.16)


It will also be useful to compute the expected present value of the contract under the un-derwriter’s discount rate given high effort and low effort.

E

[∫ ∞0

e−γtdXt|e = 1

]= K +Nα

u+ λLR

γ + λL

E

[∫ ∞0

e−γtdXt|e = 0

]= K +Nα

u+ λHR

γ + λH

An optimal contract of the form given in (2.16) must make both the participationconstraint and the incentive compatibility constraint bind. Hence we have the followingsystem of equations

a0 + C = K + αNu+ λLR

γ + λL

a0 = K + αNu+ λHR

γ + λH.

Which we can solve to get

α =C(γ + λH)(γ + λL)

N(u− γR)(λH − λL)

K = a0 −C(u+ λHR)(γ + λL)

(u− γR)(λH − λL)

The cost of providing such a contract, which we denote by bE(a0), will then be

bE(a0) = K + αNu+ λLR

r + λL

= a0 + C +

(C(γ + λH)(γ − r)


)(u+ λLR

r + λL

)We assume that investors earn zero profits and calculate the implied promised value to theunderwriter to get

aE0 (N) =

(N − C(γ + λH)(γ − r)


)(u+ λLR

r + λL

)− C.

Figure 2.2 shows the percentage gain in underwriter value for a range of values of theparameters λH and γ holding the other parameters fixed. The efficiency gain from using theoptimal contract over the alternative contract increases as either γ increases or λH decreases.As γ increases, the gains from securitization increase, and hence any improvement in theefficiency of the structure of securitization will have a larger effect on efficiency. As λHdecreases, the severity of the moral hazard problem increases since we assume the cost ofeffort stays constant. That is, effort becomes harder to infer since λH and λL are closer in


Comparing the optimal contract to a fraction of the mortgage pool

1% 1.5% 2% 2% 3%ΛH

2.5%

5%

7.5%

10%

12.5%

15%

Percentage Gain

in Underwriter Value

Γ = 5.5%

Γ = 6.5%

Γ = 7.5%

Figure 2.2: The percentage gain in underwriter value from using the optimal contract versusa contract in which the underwriter retains a fraction of the pool of mortgages. Parametervalues : r = 5%, λL = .5%, N = 100, R = (50%)u

rand c = (.25%)Nu

r


value, however the cost of effort stays the same, hence shirking becomes more attractive.As such, harsher incentives are needed to implement high effort and the optimal contractdelivers larger efficiency gains over the alternative contract. The overall level of gains is highindicating that the equity-like contract is very inefficient. In fact, for some parameter valuesthere does not exist an equity-like contract that yields positive profits to the investors andprovides incentives for effort.

2.2.2 The optimal contract versus a “first loss piece”

The next alternative contract we consider is the so called “first loss piece.” In the mostsimple form of this structure, the mortgages are pooled and two tranches are sold to investors,a senior tranche, and a junior tranche or first loss piece. The underwriter retains a largeenough junior tranche to maintain incentive compatibility. To define this contract we let Ytand Zt be the cumulative cash flow paid to the senior and junior tranches respectively bytime t. The cash flow from the mortgages is distributed to the tranches according to thefollowing rules

dYt = (N −maxn,Dt)udt+RdDt (2.17)

dZt = maxn−Dt, 0udt (2.18)

for some n < N which determines the size of the junior tranche.11 We can design an incentivecompatible contract using the above tranche structure as follows. The underwriter retainsthe junior tranche as well as receives the proceeds from the sale of the senior tranche. Wecall such contracts first loss piece contracts.

We state the value of the first loss piece and the optimal first loss piece contract in thefollowing proposition

Proposition 2. The value of the first loss piece of size n given the arbitrary discount factorδ and default intensity Λ is

F (δ,Λ, n) =u

δ

(n−

n∑k=1

N !Λk

(N − k)!

1∏k−1m=0(δ + (N −m)Λ)

)(2.19)

The optimal first loss piece contract is given by n such that

n = arg minmm : F (γ, λL,m)− F (γ, λH ,m) ≥ C (2.20)

Proof. See Appendix.

11The reader may note that the first loss piece we have defined does not have claim to the recovery valuesof the first n defaulted mortgages. In reality, the first loss piece may indeed have such a claim. In our setting,paying the recovery value to the first loss piece would reduce its efficiency, though not substantially.


The value of the first loss piece given by equation (2.19) appears quite complicated butyields a simple decomposition. Consider a claim to a constant cash flow of udt until the kthdefault. We can express the underwriter’s value for this claim as

E

[∫ τk

0

ue−γtdt|e = i

]=u

γ(1− δk)

where δk is some discount determined by the distribution of τk. We can directly calculate δkusing the distribution of the order statistics of an independent identically distributed sampleof exponential random variables with intensity λi. Summing the value of such claims fork = 1 to k = n we are left with equation (2.19).

The intuition behind the optimality of the contract in Proposition 2 is the following.As the size of the first loss piece increases, the expected life of the first loss piece increasesand hence the contract becomes less efficient. So the optimal contract features the smallestfirst loss piece such that the contract provides incentives to exert effort. We could improveon this contract by allowing the underwriter to sell a portion of the first loss piece to theinvestors, however such an improvement would be insubstantial for reasonable parameters.

Again, we can assume the investors are competitive and hence make zero profits. Figure2.3 shows the gain in underwriter value from using the optimal contract versus the first losspiece structure. For parameter values with small gains from securitization and a small moralhazard problem, the first loss piece is a very good alternative to the optimal contract. Thisfact is robust to various pool sizes N . However, when γ is large or λH is small, the gain fromusing the optimal contract is substantial, for example more 12 basis points when γ = 7.5 andλH = .525%. It should be noted that the agency problem must be very severe to generatea substantial gain in efficiency from using the optimal contract versus the first loss piececontract.

In light of Figure 2.3, we can conclude that in many instances the first loss piece is a veryefficient incentive contract. The main reason for this conclusion is that the first loss piece,as presented above, accelerates payments relative to other contracts, such as the contract weconsider in the previous section.

2.3 Extensions

2.3.1 An Initial Capital Constraint

In this section we consider the optimal contracting problem when the underwriter facesan initial capital constraint. This case arises when the underwriter does not have sufficientinternal capital to originate the mortgages. The structure of the contract remains largelyunchanged, except for the addition of a transfer at t = 0.

Suppose the underwriter requires K in initial capital at t = 0 to originate the mortgages.Specifically we add the following constraint

dX0 ≥ K (2.21)


1% 1.5% 2% 2% 3%ΛH

.025%

.05%

.075%

.10%

.125%

.15%

Percentage Gainin Underwriter Value

Γ = 5.5%

Γ = 6.5%

Γ = 7.5%

Figure 2.3: The percentage gain in underwriter value from using the optimal contract versusa contract in which the underwriter retains a fraction of the first loss piece. The slightnon-monotonicity arises due to the fact that as λH − λL increases, the size of the optimalfirst loss piece increase by increments of 1 rather than continuously. Thus as λH − λLincreases the loss in efficiency due to the fact that the first loss piece contract does not makethe incentive compatibility constraint bind changes non-monotonically. Parameter values :r = 5%, λL = .5%, N = 100, R = (50%)u

rand c = (.25%)Nu

r


to Definition 1. The following proposition states the solution to the optimal contractingproblem.

Proposition 3. Let a0 be the promised value to the underwriter net of origination costs K.The optimal contract Xt that satisfies constraint (2.21) is given by dX0 = K and Proposition1 for t > 0.

Proof. See appendix

The intuition behind Proposition 3 is essentially the same as for Proposition 1. Afterproviding the initial required capital K, the contract makes all subsequent transfers depen-dent on the realization of the first default time to provide incentives to the underwriter toexert effort. Once a long enough period has passed before the first default time, it is optimalto completely pay off the underwriter due to the difference in discount rates between theinvestors and the underwriter.

2.3.2 Maturity of the optimal contract

One attractive feature of the optimal contract is it’s relatively short maturity. Even forhigh values of K (the upfront capital required to originate the mortgages) the waiting periodrequired to provide incentives to the underwriter is short. Figure 2.4 shows t0 as a functionof λH for various levels of K holding other parameters constant. When λH is relativelyclose to λL and K is relatively large, the moral hazard problem is relatively more severe,since the inference problem is more difficult and the continuation value of the underwriteris smaller. However, even for K = 99.5N(u+λLR)

r+λLand λH − λL = .5%, the maturity of the

optimal contract is less than 2 years. When contrasted to the potentially infinitely livedmortgages of the model, this contract maturity is very short. This feature is appealing inpractice since it implies that even when facing severe moral hazard problems, investors inMBS can enforce underwriter effort with a short-lived contract. In other words, even thougha mortgage pool may last for 30 years, the underwriters position can be short-lived whilestill providing incentives to exert effort.

2.3.3 Partial Effort

The underwriter is endowed with an all-or-nothing effort technology in our model. Inother words, the agent must choose a single effort level for each mortgage. Alternatively, wecan consider a specification in which the underwriter can apply effort to some and not all ofthe mortgages. The optimal contract remains unchanged when we allow for such a deviationgiven a reasonable restriction on the cost of effort c detailed below. Specifically, supposethe underwriter can choose to apply effort to n ≤ N of the mortgages resulting in a poolof N mortgages in which n mortgages default according to an exponential distribution withparameter λL, and N − n mortgages default according to an exponential distribution withparameter λH . We will refer to such a strategy as applying partial effort. We alter notation


1% 1.5% 2% 2% 3%ΛH

6

12

18

24

30

36

Optimal Contract

Maturity

Hin MonthsL

K = 98%P

K = 99%P

K = 99.5%P

Figure 2.4: The maturity of the optimal contract t0 for a range of parameters, where Kis reported as a percentage price of the market value of the mortgage pool P = N(u+λLR)

r+λL.

Parameter values r = 5%, γ = 6% λL = .5%, N = 100, R = (50%)ur

and c = (.25%)Nur


to let a partial effort strategy be denoted e = n so that e = 0 corresponds to zero effort ande = N corresponds to full effort. The cost of applying a partial effort strategy is given byc · e. We modify Definition 1 to include incentive compatibility constraints for each possiblepartial effort strategy as follows

Definition 2. Given a promised utility a0 to the underwriter, a contract xnNn=0 thatimplements e = N is optimal if it solves the following problem

b(a0) = mindXt≥0

E

[∫ ∞0

e−γtdXt

∣∣e = n

](2.22)

such that

E

[∫ ∞0

e−γtdXt

∣∣e = m

]− c ·m ≤ E

[∫ ∞0

e−γtdXt

∣∣e = N

]− c ·N for all m (2.23)

and

a0 ≤ E

[∫ ∞0

e−γtdXt

∣∣e = N

]− c ·N. (2.24)

Proposition 4. The optimal contract is the same as that of Proposition 1.

Proof. See appendix

Proposition 4 relies on the fact that under the contract detailed in Proposition 1 andthe flexible effort technology, the underwriter does not gain more by deviating to a strategyin which she applies effort to some and not all of the mortgages than a strategy in which sheapplies zero effort. This implies that the original optimal contract satisfies the additionalincentive compatibility constraints ruling out partial effort deviations since it satisfies theoriginal incentive compatibility constraint ruling out the zero effort strategy. Moreover, thespace of contracts that satisfy (2.23) is contained in the set of contracts which satisfy (IC)since constraint (IC) is contained in (2.23). Since the contract of Proposition 1 satisfiesthe additional constraints induced by partial effort strategies and is optimal over the largerspace of contracts satisfying constraint (2.23), it is optimal over the smaller space of contractssatisfying the additional constraints.

2.3.4 Adverse selection

Throughout the above analysis, we have focused on a moral hazard setting in whichthe underwriter makes a hidden effort choice that affects the risk of the mortgages she sellsto the investors. In this section, we show how our model can be altered to address anadverse selection problem in which the underwriter is endowed with mortgages with a givendefault risk and wishes to sell them to secondary market investors. This setting is similar toother papers in the literature, notably DeMarzo (2005). Our main result is that the optimalcontract for an underwriter with low risk mortgages remains qualitatively unchanged.


In a standard adverse selection model of asset backed security design, the issuer, inour case the underwriter, has private information about the assets she wishes to sell. Wemodel this by assuming that the underwriter has N mortgages to sell, all of which are eitherlow risk or high risk where mortgage cash flows are given in section 2.1.1. We look for aseparating equilibrium in which the underwriter can signal the quality of her mortgages bychoosing a contract. Specifically, we look for a pair of contracts (XH , XL) such that anunderwriter with high risk mortgages chooses the contract XH , and an underwriter with lowrisk mortgages chooses the contract XL. As such, we can view the choice of contract as asignal of pool quality. Formally, we define a separating equilibrium as follows.

Definition 3. A separating equilibrium is a pair (XH , XL) such that:

1. The underwriter chooses XH when she has λH type mortgages:

XH ∈ arg maxX

E

[∫ ∞0

e−γtdXt|λH]

(2.25)

2. The underwriter chooses XL when she has λL type mortgages:

XL ∈ arg maxX

E

[∫ ∞0

e−γtdXt|λL]. (2.26)

3. Investors earn zero expected profits:

E

[∫ ∞0

e−rtdXHt |XH

]=N(u+RλH)

r + λH, (2.27)

E

[∫ ∞0

e−rtdXLt |XL

]=N(u+RλL)

r + λL. (2.28)

Since all gains from securitization arise by front loading payment to the underwriter,and investors are competitive and earn zero profits, we choose

dXHt =

N(u+λH)r+λH

for t = 0

0 otherwise. (2.29)

Of course separating equilibria with different XH may exist, however, such equilibria will bePareto dominated by equilibria with XH described by equation (2.29).

In its current form, Definition 3 is not directly related to the contracting problem givenin Definition 1. However we can draw a relation between the two definitions by findingthe least cost sperating equilibrium, which is the equilibrium which satisfies Definition 3 andmaximizes the expected payment to the underwriter when she has type λL mortgages. The


least cost separating equilibrium thus solves the following problem:

maxX

E

[∫ ∞0

e−γtdXt|λL]

(2.30)

s.t.E

[∫ ∞0

e−γtdXt|λH]≤ N(u+ λH)

r + λH,

E

[∫ ∞0

e−rtdXt|λL]≤ N(u+ λL)

r + λL.

Problem (2.30) is the dual problem of (I) and thus we can apply Proposition 1 to get thefollowing proposition.

Proposition 5. The least cost separating equilibrium is given by dXH0 = N u+λHR

r+λH= aH0 and

dXHt = 0 for t > 0, and

dXLt =

0 if t 6= t0,y0I(t0 ≤ τ1) if t = t0,

where

t0 =1

N(λH − λL)

aL0aH0

y0 = e(γ+NλL)t0aL0 ,

where

aL0 =

(Nu+ λHR

r + λH

)1/(1+η)

(aH0 )η/(1+η) (2.31)

and

η =γ − r

N(λH − λL). (2.32)

Proof. Follows directly from Proposition 1.

In a static framework, such as that of DeMarzo (2005), the signal space is limited to thefraction the underwriter chooses to retain of some security backed by the mortgages. In oursetting, the signal space is much richer since it includes any payment profile through timethat is adapted to the information filtration generated by the cumulative default processof mortgages. Accordingly, the most efficient signal in our model uses timing as a centralfeature rather than the fraction of some high risk tranche retained by the underwriter.

2.4 Conclusion

This paper studies a model of mortgage securitization in a moral hazard setting thathighlights an important aspect of contracting in mortgage markets, namely that information


is revealed over time. We find that the optimal contract is a lump sum payment from theinvestors to the underwriter conditional on a period of no defaults. In addition, we evaluatetwo alternative contracts and find that a “first loss piece” style contract comes very close toachieving the same level of efficiency as the optimal contract.

If we view the contracting problem as essentially a hypothesis testing problem, a naturaltrade-off arises between the power of the test used in the contract and the amount of timeneeded to implement the test. This intuition gives rise to three new findings. First, the timingof payments to the underwriter is a key mechanism providing incentive to the mortgageunderwriter to exert effort. Second, the optimal contract maturity can be short while themortgages are long lived. Finally, mortgage pooling, the process whereby mortgages arebundled to create a collateral pool, follows from an information enhancement effect : byconditioning all payments on an aggregate signal taken from the entire pool of mortgages,rather than observing each mortgage individually, the optimal contract achieves the bestpossible trade-off between the testing power and the testing time.

One could raise a concern that the optimal contract provides incentives to both partiesto manipulate the default process. For example the underwriter could have an incentive topostpone default times until she receives payment. Similarly, investors could have an incen-tive to bribe a single borrower to default early in order to avoid payments to the underwriter.One possible solution to this problem would be to use a third party servicer, thereby limitingdirect access to borrowers rendering any such manipulation more costly. In addition, directlyaffecting default times through side payments would constitute fraud, making manipulationsubject to significant legal risk. More importantly, we note that in the presence of moderatemanipulation costs, the first loss piece, which closely approximates the optimal contract,does not create incentives for the investors or underwriter to manipulate mortgage defaultsbecause a single mortgage default has only limited contractual consequences. In sum, al-though manipulation of the default process could be a concern, we believe it is not of firstorder importance since simple legal and practical institutions can prevent it with little lossof efficiency.

We consider three important extensions to the basic model. The main result is that thestructure of the optimal contract is robust to altering the contracting problem in plausibleways. The first extension introduces an initial capital constraint, that is the underwriterlacks sufficient initial capital to originate the mortgages. The qualitative features of thecontract remain unchanged after introducing this constraint. The only significant differencebeing a time zero transfer from the investors to the underwriter exactly equal to the capitalrequired to originate the mortgages. After time zero, the optimal contract is identical to theoptimal contract without the initial capital constraint except for the magnitude and timingof the one time lump sum transfer.

The next extension we consider is a flexible effort technology. It is possible that theunderwriter could apply costly underwriting practices to some and not all of the mortgages.In this case, we identify a set of additional incentive compatibility constraints induced by thenew effort technology. We then show, that the original optimal contract satisfies these newconstraints and hence solves the optimal contracting problem of the more restricted space


of incentive compatible contracts.In the third extension, we show how our result can be adapted to an adverse selection

setting. In this setting, we view the choice of contract structure as a signal of the under-writer’s private information. The problem then becomes finding the most efficient signal,and can easily be mapped into our original contracting problem.

The extensions we consider are certainly not exhaustive. For example, we could haveconsidered risk averse underwriters or investors, correlation among mortgage defaults, timevarying default rates, or a richer action set for the underwriter. These additional featureswould potentially change the specific lump sum form of the optimal contract and are allpotential directions for future research. For example, time varying default rates may extendthe duration of the optimal contract while a richer action set may provide insights intothe equilibrium level of risk in mortgage pools by allowing for interior levels of effort to beoptimal. However, the key economic forces of the model, specifically the relative impatienceof the underwriter and the dynamic nature of information revelation, will remain unchangedand the optimal contract will balance front loading payments with providing incentives toexert effort.

28

Chapter 3

Reputation and signaling in asset sales

3.1 Introduction

In many important financial markets, issuer private information leads to a basic adverseselection problem. For example, an issuer of asset backed securities (ABS) may know moreabout underlying assets than do investors. In practice, such an issuer can reveal her privateinformation either by retaining a fraction of the securities she issues (costly signaling) or bymaintaining a reputation for honesty. Although there is a substantial literature investigatingcostly signaling and one investigating reputation, there is little work that considers thepossible interactive effects of these two mechanisms. This paper presents a model of securityissuance under asymmetric information that allows for both costly signaling and reputationeffects. I consider the problem faced by an issuer selling assets to investors in a repeatedgame. In a given period, the issuer is endowed with a single asset. She perfectly observesthe quality of this asset while potential investors do not. This asymmetry of informationmeans that the issuer faces a lemons problem as in Akerlof (1970). The issuer may signalasset quality by retaining a portion of the asset. At the same time, the issuer may build areputation for truthfully reporting asset quality through the performance of past assets.

Combining costly signaling and reputation admits three new findings. First, issuerretention can decrease with issuer reputation, indicating that costly signaling and reputationcan act as substitutes. Second, the equilibrium relationship between retention and issuerreputation implies that a better reputation can decrease the probability that the issuer willtruthfully reveal asset quality. Finally, equilibrium prices can be U-shaped in reputation.These results translate into an important empirical implication for ABS markets. The assetof the model can be thought of as the informationally sensitive portion of a securitization.Under this interpretation, the first result of the model implies that an ABS issuer with aworse track record will retain more of given issue. I test this implication of the model usingdata on subordination levels in the commercial mortgage backed securities (CMBS) marketand find that issuers who have experienced a greater number of downgrades on past dealswill retain more of the below investment grade principal of new deals.

Chapter 3. Reputation and signaling in asset sales 29

The model is an infinite repetition of a securitization stage game. In each stage game, arisk neutral issuer is endowed with an asset to securitize for sale to investors. Nature chooseswhether the asset is the good type or the bad type, where asset type denotes expectedfuture cash flow. The asset yields cash flow one period after it is securitized. The cash flowdistribution is intentionally simple in order to abstract from security design problems. Thereis a competitive market in which relatively patient risk neutral investors will buy a fractionof the asset. The issuer can perfectly observe the type of the asset, however this informationis hidden from investors and non-verifiable at the time of securitization. The issuer mayreport or misreport the type of the asset to the investors in a prospectus. In addition, theissuer can signal asset type by retaining a fraction of the asset. Because the issuer has ahigher discount rate than investors, a common assumption in the literature, such a signal iscostly and hence credible.

Reputation concerns arise due to asymmetric information over issuer preferences forhonesty. Specifically, the issuer could be of two possible types. The honest type issueris committed to truthfully reporting the asset type in the prospectus. In contrast, theopportunistic type issuer will choose her reporting strategy by maximizing her payoffs. Bothtypes optimally choose a fraction to retain. The issuer’s reputation is the probability theinvestors place on the issuer being the honest type. By mimicking an honest issuer, i.e.truthfully reporting asset type in the prospectus, an opportunistic issuer can improve herreputation and thereby reduce the lemons discount on the fraction of the asset sold toinvestors.

Issuer type in the model can be viewed as a proxy for the issuer’s preferences over accu-racy. For example, the issuer may have separate lines of business that depend on reputationin an opaque fashion. This would be the case for an investment bank with many lines ofbusiness, all of which depend on it’s reputation for accuracy. Investors in one type of productissued by this bank, say CMBS, may not know the profitably and sensitivity to reputationof the bank’s underwriting business. A bank with a highly profitable underwriting busi-ness would correspond to the honest type, while a bank with a less profitable underwritingbusiness would correspond to the opportunistic type.

Combining costly signaling and reputation leads to multiple equilibria. The simplestclass of equilibria, which I call separating equilibria, arises when the issuer perfectly revealsthe quality of assets through either retention or public report. In a separating equilbrium,the ex post performance of an asset does not yield any new information about the type ofthe issuer, so reputation and price remain constant. A truth telling equilibrium, a specialtype of separating equilibrium, obtains when the issuer’s public report is credible regardlessof issuer retention. I show that a truth telling equilibrium exists if and only if the issuer issufficiently patient.

The next class of equilibria, which I call mixed strategy equilibria, arises when theopportunistic issuer deviates from truthful reporting at least part of the time and does notperfectly reveal asset quality through retention. For certain parameter values, this typeof equilibrium will Pareto dominate the separating equilibrium. In the discussion below, Icharacterize a particular mixed strategy equilibrium. In such an equilibrium, the issuer does


not perfectly reveal asset type and the reputation of the issuer fluctuates according to theex post asset performance. Retention now becomes a signal of the opportunistic issuer’sreporting strategy. The importance of retention necessarily varies with the reputation ofthe issuer since as issuer reputation increases, the strategy of the opportunistic issuer has alower impact on price.

The mixed strategy equilibrium delivers a theoretical link between issuer reputation, andasset retention. As might be expected, retention decreases with issuer reputation. What isless obvious is that the opportunistic issuer will decrease the probability that she will truth-fully report a bad type asset as she gains a better reputation. Ultimately, the opportunisticissuer will “cash in” on her reputation by reporting that a bad type asset is the good typeand collect one period payoffs. This will occur even when the issuer retains a positive frac-tion of the asset. In addition, the mixed strategy equilibrium demonstrates that the pricefor reportedly good type assets may not depend monotonically on issuer reputation. Forlow levels of reputation, the issuer perfectly reveals asset type through retention alone, andprices will be equal to the full information case. For higher levels of reputation, the issuer’spublic report is more credible and prices for reportedly good type assets will be close to thefull information value of a good type asset regardless of retention.

Finally, the model shows that although the addition of reputation to a costly signalingframework can increase issuer payoffs, it does not increase the probability of perfect informa-tion transfer. This result casts an interesting light on the claim that reputation is the coreself-disciplining mechanism for ABS markets. If one ignores the possibility that issuers maysignal asset quality through costly retention, reputation certainly provides some incentivesfor issuers of ABS to truthfully reveal their private information. However, including costlyretention shows that reputation may actually diminish the issuer’s incentives to truthfullyreveal information. This feature of the model is appealing given that opportunistic behaviorallegedly occurred in many ABS markets.

This chapter relates to the long literature studying the role of private information incausing distortions in markets Akerlof (1970). As early as Myers and Majluf (1984), thisidea has been applied to financial markets. If entrepreneurs know more about investmentopportunities than outside investors, then the irrelevance of capital structure Modigliani andMiller (1958) no longer holds and a particular security design may be more advantageous thananother. This effect leads to the “pecking order” theory of capital structure, with mangersissuing the least informationally sensitive securities (i.e. debt) first. Using a similar model toMyers and Majluf (1984), Nachman and Noe (1994) show that debt is the optimal securitydesign to finance investment over a very broad set of payoff distributions. For the specialcase of asset backed securities, Riddiough (1997b) shows that a senior-subordinated securitystructure dominates whole asset sales when an issuer has valuable information about assetsand liquidation motives are non-verifiable. However, the pecking order literature assumesan investment of fixed size, resulting in a pooling equilibrium in which no mechanism ofinformation transimission exists between issuers and investors.

Another branch of the literature focuses on signaling mechansims in security design.Leland and Pyle (1977) introduce the notion that retention can be a credible signal of private


information because it is costly, in their case because of reduced risk sharing. DeMarzo andDuffie (1999) build on this signaling mechanism and show how debt arises optimally bycreating a more informationally sensitive security which the issuer can retain in order tosignal her private information. The addition of a signaling mechanism leads to a separatingequilibrium in which all information costs are borne through signaling rather than throughprices as in pooling equilibria. DeMarzo (2005) uses the signaling theory of security design,to understand the benefits of pooling and tranching in the market for asset backed securities.Downing, Jaffee, and Wallace (2009) consider the case when issuers cannot choose partialretention and a market for lemons ensues. Unlike the previous literature on signaling andsecurity design, the model I consider focuses on binary cash flow distributions for the sake oftractability. However, the key difference between this paper and previous work on signalingequilibria in security design is that I include dynamic reputation effects. As a result, issuersmay choose to do some signaling, but such signaling will not lead to a perfect separatingequilibrium all of the time. In that sense, this paper provides a theoretical rational for amiddle ground between separating and pooling equilibria.

The second literature to which this paper relates is that of reputation affects in repeatedgames. Intuitively, agents involved in repeated games may try to attain a reputation for acertain characteristic in early stages of the game if that characteristic improves payoffs in laterstages. Kreps and Wilson (1982) and Milgrom and Roberts (1982) introduce the notion thatimperfect or asymmetric information about player preferences can provide such a mechanism.By observing a given player’s previous actions, other agents can form beliefs about thatplayer’s type. When a player may be one of two types, e.g. honest or opportunistic, sucha learning process provides a means by which that player can gain a reputation. If havinga reputation for being honest leads to higher equilibrium payoffs, then an opportunisticplayer may employ the equilibrium strategies of the honest type. Thus, reputation canact as a effective mechanism to encourage “desirable” characteristics. This literature oftenassumes that the desirable type plays a mechanical strategy and does not optimize. Inthe model below, I depart from this assumption by allowing the honest type to optimizeover her retention strategy. To my knowledge, this is the first model to include a separatedimension of the strategy space over which the honest type player has as much flexibility asthe opportunistic type player.

Some notable papers have applied the Kreps and Wilson (1982) concept of reputationto financial markets. Diamond (1989) shows that the possibility of acquiring a reputation forgood investment opportunities can incentivize firms to choose safer investments by loweringthe cost of borrowing for firms with good reputations. Diamond (1991) examines the choicebetween bank and public debt in the presence of reputation. John and Nachman (1985)analyze the role of reputation in mitigating the problem of underinvestment induced byrisky debt in the presence of asymmetric information. They show that a reputation for“good” investment policies can lead to higher prices in bond markets, and hence firms willwant to implement such an investment policy to attain a good reputation. Benabou andLaroque (1992) show that when private information is not ex post verifiable, insiders havean incentive to exploit a reputation for honesty in order to manipulate securities markets


for gain. These papers focus the efficacy of reputation to implement “good” behavior infinancial markets without considering other mechanisms that may have similar effects. Incontrast, I examine the interaction of reputation with costly signaling.

A very closely related paper is Mathis, McAndrews, and Rochet (2009), here after MMR,which considers a reputation building model of credit rating agencies. They show that theeffectiveness of reputation concerns for imposing market discipline on rating agencies dependsimportantly on the parameters of the model. Under some circumstances, reputation maybe enough to impose complete honesty on rating agencies. Although some aspects of mymodel resemble that of MMR, unlike MMR I allow the issuer to retain a portion of anysecurity she issues, creating an additional mechanism for the credible revelation of privateinformation. MMR only allow the stage game payoff to depend on reputation through asingle equilibrium strategy: the probability the rating agency tells the truth. While ratingagencies do not typically retain a meaningful stake in an issue that they have rated, securitiesissuers do. In this way, my model explores the interaction of reputation effects with a moreformal mechanism like costly retention, an interaction which is understudied in the literature.

There is also an empirical literature which examines the relationship between issuerreputation and retention. Lin and Paravisini (2010) find the lead arrangers in the syndicatedloan market make larger capital contributions (i.e. retain) more after exposed involvementin the accounting scandals of the early 2000’s. In this case, issuer (lead arranger) retention(capital contribution) is likely used to mitigate a moral hazard, rather than adverse selection,problem. Sufi (2007) finds a similar relationship between lead arranger reputation andcapital contribution by using market share as a measure of reputation. He shows thatlead arranger capital contribution weakly decreases in lead arranger market share althoughcapital contribution never reaches zero. This relationship is likely do to a combinationof moral hazard and adverse selection problems. Finally, Titman and Tsyplakov (2010),who investigate the link between mortgage originator performance, commercial mortgagespreads and default rates, and CMBS structure. They find originator’s with poor stock priceperformance originate lower quality mortgages which also end up in CMBS deals with moresubordination (less AAA rated principal).

3.2 The model

3.2.1 Assets, agents and actions

The economy is populated by an issuer and a measure of competitive investors. Theissuer has the constant per period discount factor γ < 1 and the investors have the perperiod discount factor 1. The difference in the discount rates of the issuer and investorsrepresents the relative impatience of the issuer and drives the gains from trade in the model.The relative impatience of the issuer could arise for a variety of reasons, including capitalrequirements and access to additional investment opportunities.

Time is infinite and indexed by t. At fixed dates t = 0, 1, 2, . . . the issuer is endowed


with a single asset which produces a cash flow Xt+1 at the beginning of the next date andis one of two possible types. For convenience, I denote the type of the asset at time t bythe process At ∈ G,B. The asset is the good type with probability λ and is the bad typeotherwise. Good assets produce a cash flow of 1 while bad assets produce a cash flow ` suchthat 0 < ` < 1.1 These asset cash flows imply perfect observability since after observingcash flows the type of the asset is known. Perfect observability will allow me to explicitlycharacterize strategies and value functions.

At the start of date t, the issuer may sell a fraction qt ∈ [0, 1] of the asset to investors.The investors observe the quantity qt at each date. The issuer has an incentive to sell theasset since investors are relatively patient. In practice, it may be impossible for the issuerto choose q ∈ (0, 1) due to regulatory restrictions as considered in Downing, Jaffee, andWallace (2009). I consider this type of restricted signaling space as an extension to the basicmodel. In addition to choosing the level of retention, the issuer produces a prospectus whichcontains a report at ∈ g, b indicating the type of the asset.

3.2.2 Issuer type, reputation and strategies

Following Kreps and Wilson (1982) reputation arises from incomplete information aboutissuer preferences. Specifically, the issuer can be of two possible types. The honest type issueralways provides a truthful report. Moreover, she chooses a quantity of the asset to issue thatmaximizes her expected proceeds from securitization and retained assets. In contrast, theopportunistic type issuer chooses both a report and quantity to maximize her expectedproceeds from securitization and retained assets. The formal definition of the objectivefunctions of both types of issuer is given in Definition 4. The reputation of the issuer is thensummarized by the probability the investors place on the issuer being the honest type withan initial probability φ0. As time evolves, the investors update their beliefs about issuertype by observing the history of public information of the game, denoted Ht. The investors’belief that the issuer is the honest type is thus given by

φt = P(issuer is honest type|Ht).

I will assume the current quantity Q and the reputation φt contain all the relevantinformation for the beliefs of the investor about the current asset type given a public report.That is, I assume that φt is a Markov state variable for the history of the game. In principal,the investors’ beliefs about the current asset type could depend on the entire history ofthe game and in particular the path of past quantities. Because such a dependence makesthe notation overly cumbersome, I do not consider it in the main text. In Appendix ??, Iallow investor beliefs to depend on the path of past quantities and reports and show that

1Since the focus of the present paper is to jointly analyze the effect of costly signaling and reputation, Iconsider binary asset cash flows and abstract away from security design issues. For an analysis of securitydesign under a rich set of cash flow distributions in a static setting see Nachman and Noe (1994) or DeMarzoand Duffie (1999).


restricting attention to investor beliefs which are Markov in reputation does not rule outimportant equilibria.

The market must price the asset based on the report of asset type, the issued quantity,and the reputation of the issuer. If the issuer reveals that the asset is the bad type, theninvestors believe the asset is the bad type with probability one, so the market will pay a price` per unit for the offering. When the issuer reports that the asset is the good type, the marketprice per unit is given by the inverse demand curve, denoted P (q, φ) : [0, 1]× [0, 1]→ [0, 1],which is the value investors place on the underlying asset of an ABS with quantity q andreport g offered by an issuer with reputation φ. In other words, if an issuer with reputationφ offers an ABS with quantity q and report g, she will receive qP (q, φ) in proceeds fromthe sale of the ABS to investors. It is potentially costly for the issuer to choose q < 1because she has a larger discount rate than the investors. Accordingly, the level of retentionrepresents a credible signaling mechanism for an issuer with a good type asset. That is,investors should believe that an ABS with a relatively higher level of retention is backedby an asset of relatively higher quality. Thus, for each φ, the inverse demand curve P (q, φ)is downward sloping in q. This setup is a simplified version of the signaling mechanism ofDeMarzo and Duffie (1999), with the exception that the demand curve may now depend ona report of asset quality and issuer reputation.

Both issuer types form strategies conditional on their current reputation and the typeof the current asset. Before formally defining the issuer’s strategy space, I make someconvenient assumptions to simplify notation. Specifically, I assume that either type of issuerwill always provide a truthful report when the asset is the good type and will always sellthe entire asset when she reveals that the asset is the bad type.2 Thus a reporting strategyis a function π(φ) : [0, 1]→ [0, 1] giving the probability of accurately reporting a bad asset’strue type. A quantity strategy is a function Q(φ) : [0, 1] → [0, 1] giving the fraction of theasset sold to investors when the issuer reports that the asset is the good type. Recall thatthe set of admissible strategies for the issuer depends on her type. The set of admissiblestrategies for the opportunistic type, denoted AO, is simply the set of all possible strategiespairs defined above, whereas the set of admissible strategies for the honest type issuer isgiven by AH = (π,Q) ∈ AO|π(φ) = 1. The restriction that defines AH reflects the factthat the honest type is committed to truthfully revealing a bad type asset.

To recapitulate, the timing of the game is as follows. At each date t = 0, 1, 2 . . . theinvestors and issuer play a securitization stage game. At the beginning of a given date t,the cash flow from the asset sold on the previous date is realized and the players updatethe reputation of the issuer. Second, the current asset type is revealed to the issuer and shechooses her report and retention strategies. Finally, investors buy the security at a priceqP (q, φ) and the process starts anew. Figure 3.1 gives a time-line of the game with the

2This assumption is without loss of generality. The opportunistic type issuer would never report thata good type asset is the bad type in equilibrium since doing so would not increase her reputation or herinstantaneous proceeds from asset sale. Thus, if the issuer reveals that the asset is the bad type, then herprivate value for the asset is always less than that of the investors, regardless of her quantity strategy, andselling the entire asset is optimal.


sequence of actions that occur within a given date t.

3.2.3 Equilibrium

At any given date t the issuer will maximize the discounted expected value of herproceeds from securitization plus the cash flow from retained assets. Formally an issuer withreputation φ has an instantaneous cash flow to an action (q, π) given by

Ut(q, π, P, φ) = λ(qP + γ(1− q)) + (1− λ) [π`+ (1− π)(qP + γ`(1− q))] , (3.1)

when facing the demand curve P at time t.

Definition 4. The quadruple (P,QH , π,QO) is an equilibrium if at all times t

1. The strategy of the honest type maximizes her payoffs:

QH ∈ arg maxqEt [∑∞

n=t γn−tUn(qn, 1, P, φn)],

2. The strategy of the opportunistic type maximizes her payoffs:

(QO, π) ∈ arg maxq,π Et [∑∞

n=t γn−tUn(qn, πn, P, φn)],

3. φt is determined using Bayes rule whenever possible, and

4. Investors earn zero expected profits: P (Qi(φt), φt) = E [Xt+1|φt, Qi(φt)] for i ∈ O,H.

5. An equilibrium is separating if P (QH(φt), φt) = P (QO(φt), φt) = 1.

To be clear, by using the term separating equilibrium, I am referring to equilibria thatreveal the true type of the underlying assets, rather than the true type of the issuer. Indeed,it is impossible for the issuer to credibly reveal her type via a particular retention strategy inequilibrium as will become apparent shortly. I will often refer to a given equilibrium as theleast cost separating equilibrium. By this, I mean the separating equilibrium which deliversthe highest payoff to the issuer.

3.3 Solution

3.3.1 The game without reputation

To begin the analysis, I consider equilibria when the issuer is known to be the oppor-tunistic type. When the issuer is revealed to be the opportunistic type by the history ofthe game, so that φt = 0, Bayes Rule implies that φs = 0 for all s ≥ t. In other words,a reputation of zero is an absorbing state. Thus, equilibria in this state will serve as animportant input to the solution of the general case.

Before considering the repeated game, it is useful to consider equilibria of the staticgame. The natural restriction of Definition 4 for the static game replaces conditions (1)and (2) with a one period maximization problem. The following proposition summarizes theequilibria of the static game without reputation.


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• A separating equilibrium is given by Q = q, π = 1, and

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• A pooling equilibrium exists and is given by Q = 1, π = 0, and P (q, 0) = λ+ (1− λ)`for all q ∈ [0, 1] if and only if γ ≤ λ+ (1− λ)`.

Other equilibria, in particular those with mixed reporting strategiess in which 0 < π < 1,may also exist. However, as will become clear when considering repeated versions of thestatic equilibria, mixed strategy equilibria of the static game without reputation are Paretodominated by the least cost separating equilibrium or the pooling equilibrium, dependingthe parameterization of the model.

The least cost separating equilibrium follows from the classic signaling intuition. Thequantity q is defined so that even when the market responds with a price per share of onefor the quantity q, the issuer with a bad type asset is better off selling the entire asset fora price of `. At the same time, the issuer with a good type asset strictly perfers selling thequantity q at a price per unit of one to retaining the entire asset. Such a quantity q existsbecause the relative impatience of the issuer implies that retaining a fraction of the asset ismore costly for the issuer when she has a good type asset than when she has a bad typeasset since the issuer is less patient than the investors.

The pooling equilibrium arises when the issuer’s value for retaining a good type assetis less than the ex ante expected value of the asset to the investors. In a standard staticsignaling game, pooling equilibria are typically ruled out by the D1 refinement of Cho andKreps (1987) which restricts off equilibrium beliefs. However, it is not clearly valid to applythis refinement to the repeated game. At the same time, the pooling equilibrium of the staticgame only exists for parameterizations of the model in which the lemons problem is not too“severe” and markets could function efficiently without any means of information transfer.Specifically, a repetition of the pooling equilibrium would lead to payoffs equal to the fullinformation case. For the remainder of the paper I will assume that parameters are suchthat the pooling equilibrium does not exist:

Assumption 1. The parameters of the model do not admit a pooling equilibrium of thestatic game: λ+ (1− λ)` ≤ γ.

I can now consider equilibria of the repeated game for φt = 0. Since investors’ strategieswere assumed to be Markovian in φ, there is no mechanism to make the issuer’s currentpayoffs depend on her past actions. And thus, public reports of asset quality are no morecredible than in the static version of the game. This observation leads to the followingproposition.


Proposition 7. Suppose φt = 0. Then φs = 0 for all s ≥ t, and a strategy pair (Q, π) andprice schedule P (q, 0) are an equilibrium if and only if they are an equilibrium of the staticgame.

In principal, investors could play punishment strategies in which past quantities affectbeliefs about current asset types even though reputation is fixed at zero. The result ofsuch a strategy would be that a truth telling equilibrium may emerge even though φ0 = 0.This type of equilibrium behavior is sometimes thought of as resulting from “reputation”,however this is not the concept of reputation I consider here. For completeness, I considerthe possibility of punishment strategies in Appendix ??. With punishment strategies, the setof possible equilibria for the repeated game with no reputation would include a truth tellingequilibrium, provided the parameters satisfy a given restriction. It will turn out that thisrestriction is also a necessary condition for the existence of a truth telling equilibrium for thecase with positive reputation. Thus, by assuming away punishment strategies, I have onlyeliminated truth telling equilibrium for the no reputation state. In particular, introducingpunishment strategies does not allow for additional truth telling equilibria for the positivereputation state.

Propositions 6 and 7 imply there are mulitple equilibria for the repeated game withoutreputation. Since the structure of equilibria with positive reputation will hinge on whatequilibrium strategies obtain if the issuer’s reputation falls to zero, it is neccessary to have aconsistent means of selecting an equilibrium in this state. Again, applying the D1 refinementof Cho and Kreps (1987) is not clearly applicable in the repeated setting. Instead, I rely onthe fact that conditional on parameters, a single equilibrium delivers the most value to theissuer. Since investors are competitve and always earn zero profits in expectation, such anequilibrium is Pareto dominant. Under Assumption 1, the least cost separating equilibriumof Proposition 6 delivers the highest equilibrium payoffs to the issuer. Thus, I will assumethe least cost separating equilibrium obtains for the no reputation state.

3.3.2 Reputation dynamics and optimization

Now that I have fixed equilibrium for the repeated game in the no reputation state, I canproceed to solve for the dynamics of reputation in equilibrium. First, I make the followingimportant observation.

Lemma 1. The honest issuer and the opportunistic issuer always issue the same quantity,QH(φ) = QO(φ) for all φ.

The intuition behind Lemma 1 is the following. The opportunistic issuer and the honestissuer both value instantaneous payoffs from the securitization of a good asset identically.Moreover, the opportunistic issuer values a higher reputation weakly more than the honestissuer. Therefore, any quantity strategy which increases reputation will be at least as attrac-tive to the opportunistic type as the honest type. This implies that the quantity issued isnot a credible signal of issuer type and cannot contain any new information about the type


of the issuer. In particular, this means that reputation is only updated during the reputationupdating phase, and not during the securitization phase. This will be a useful fact in theanalysis since it means that the issuer need not take into account the effect of her quantitystrategy on her future reputation. In addition it will allow me to simplify the analysis of thereputation updating process.

Given Lemma 1, it is straightforward to derive the dynamics of reputation in terms ofthe report and ex post performance of the asset. Let f : g, b×`, 1× [0, 1]→ [0, 1] denotethe reputation updating function. Using Bayes rule whenever possible, I have

f(g, 1, φ) = φS = φ (3.2)

f(g, `, φ) = φF = 0 (3.3)

f(b, `, φ) = φB =φ

φ+ π(1− φ). (3.4)

The optimization problem faced by the issuer can now be simplified given the reputa-tion updating function and the fact that the opportunistic and honest type issuers alwayschoose the same retention strategy. Since Lemma 1 implies that the honest type issuer andopportunistic type issuer play the same retention strategy, I drop the superscript and refer toa retention strategy as simply Q. Consider the opportunistic issuer’s problem. Let V (φ|P )denote the value function of the opportunistic type issuer when facing the demand scheduleP and VG(φ|P ) and VB(φ|P ) denote the value functions when the opportunistic type issuerfaces the demand schedule P and is endowed with good asset or a bad asset respectively.Then V (φ|P ) = λVG(φ|P ) + (1−λ)VB(φ|P ), and VG(φ|P ) and VB(φ|P ) satisfy the followingsystem of Bellman equations

VG(φ|P ) = max(π,Q)∈AO

(γ(1−Q) +QP (Q, φ) + γV (φS|P )

. (3.5)

VB(φ|P ) = max(π,Q)∈AO

π(`+ γV (φB)) + (1− π)(γ(1−Q)`+QP (Q, φ) + γV (0))

. (3.6)

3.3.3 Separating equilibria

For an equilibrium to be separating, it must be the case that by observing the quantityissued and the report given, the investors can perfectly infer the type of the asset. In general,such perfect inference can arise either because the quantity issued maps perfectly to the typeof the asset, or the loss of continuation value from being exposed as the opportunistic typeis so great that the issuer will never misreport a bad type asset. I refer to equilibria of thislatter type as truth telling. Specifically an equilibrium is truth telling if Q(φ) > q for someφ > 0 and π(φ) = 1 for all φ > 0. The truth telling equilibrium is desirable in that it allowsfor the credible revelation of issuer private information with less issuer retention. Indeed,when a truth telling equilibrium exists with Q(φ) = 1 for all φ > 0, it delivers payoffs to theissuer equal to what she would receive in a first best setting. However, some restriction onparameters is needed for a truth telling equilibrium to obtain.


Proposition 8 (Folk Theorem). Suppose φ0 > 0. There exists a truth telling equilibrium ifand only if γ ≥ 1

λ+`.

The restriction on parameters required for the existence of a truth telling equilibriumdepends on the instantaneous gains to the issuer from misreporting a bad type asset andthe loss in continuation value from being identified as the opportunistic type. Suppose theinvestors always believe the report of the issuer, then an opportunistic issuer may receive aprice of 1 for a bad type asset for one period and then be known to be the opportunistictype thereafter. Such a deviation is profitable if and only if

1− `︸︷︷︸gain in proceeds

≤ γλ(1− q)(1− γ)

1− γ.︸︷︷︸

loss in continuation value

(3.7)

This restriction simplifies to γ ≥ 1λ+`

. One interpretation of this restriction is that the issuermust be sufficiently patient so as to make a loss in continuation value severe enough toprovide incentives to always accurately report a bad type asset. In this way, the conditionsguaranteeing the existence of a truth telling equilibrium are similar to a classic folk theorem.3

When the issuer is impatient, the truth telling equilibrium can not be supported evenwith positive initial reputation. However, the repeated version of the least cost separatingequilibrium of the game without reputation still obtains.

Proposition 9. The least cost separating equilibrium of Proposition 6 where P (q, φ) =P (q, 0) for all φ is an equilibrium of the game for all φ0 ∈ [0, 1).

The cost to the issuer to credibly reveal her information about the quality of the assetto the investors thus depends importantly on the parameters of the model. When the issueris relatively patient, or 1 − λ` > λγ, she can credibly reveal her information via her publicreport without cost, so long as she has strictly positive reputation. This is the truth tellingequilibrium. If the issuer is relatively impatient, she can still credibly reveal her information,however, doing so requires her to issue a quantity strictly less than one, which is costly.Figure 3.2 shows a partition of the parameter space highlighting the region for which truthtelling is supported in equilibrium. Region I corresponds to parameters for which the truthtelling equilibrium are not supported, whereas in Region II, truth telling is supported. Anatural question is whether higher equilibrium payoffs for the issuer may be supported inRegion I by considering mixed reporting strategies. In other words, is there an equilibriumfor parameters in Region I in which the issuer achieves higher payoffs and does not alwaysperfectly reveal the type of the asset. I consider this possibility in the next subsection.

3Here the assumption that φ is a Markov state variable has some bite. If investor beliefs could dependon the path of past signals, truth telling would be supportable in equilibrium without positive reputation.Moreover the parameter restriction required would be slightly weaker.


Region I

1

1

γ

γ

Region III

`

λ

Region

II

Figure 3.2: A partition of the parameter space of the model. Region I corresponds toparameters for which the truth telling equilibrium is not supported. Region II correspondsto parameters for which the truth telling equilibrium is supported. Region III correspondsto parameters for which Assumption 1 fails and hence is not considered.


3.3.4 Mixed strategy equilibria

In light of results of the previous subsection, I look for equilibria in which 0 < π(φ) < 1for some φ ∈ [0, 1], or mixed strategy equilibria for parameters under which truth tellingis not supported in equilibrium. Specifically, I impose the following assumption to rule outtruth telling.

Assumption 2. The parameters of the model do not support the truth telling equilibrium:γ < 1

λ+`.

In what follows, I will outline a procedure to construct a particular equilibrium, thedetails of which can be found in Appendix ??. The approach will be to assume the existenceof an equilibrium in which π(0) = 1 and π(1) = 0. In other words, the opportunistic issuerwill always be honest for low levels of reputation and always submit inaccurate reports forhigh levels of reputation. I also assume a particular functional form for the demand scheduleP (q, φ). Doing so amounts to making assumptions about the off-equilibrium beliefs of theinvestors. Given, a demand schedule P (q, φ), I find Q(φ), the optimal quantity chosenby the issuer when faced with a good type asset. Next, I suppose the opportunistic issuerchooses the same quantity when faced with a bad asset for high enough levels of reputation, anecessary condition for the equilibrium to not be a separating equilibrium. Using the one-shotdeviation principal for infinitely repeated games, I find an inequality relating the quantitystrategy Q(φ) to the reporting strategy π(φ) and the value function of the opportunistictype issuer V . Finally, I construct the value function V using this inequality.

To begin the analysis, I assume a candidate equilibrium demand curve P (q, φ) is a stepfunction of q of the following form

P (q, φ) =

1 q ≤ qp∗(φ) q < q ≤ q∗(φ)` q > q∗(φ)

(3.8)

where p∗(φ) and q∗(φ) are continuous in φ such that p∗(0) = 1 and q∗(0) = q.4 The inversedemand curve P , depicted in Figure 3.3, is consistent with the investor beliefs that only anissuer with a good type asset would ever offer a quantity q less than q, while an issuer witheither type asset might offer a quantity q greater than q but less than some level q∗(φ), andonly an issuer with a bad type asset would choose a quantity q greater than q∗(φ). Given thedemand curve P (q, φ), an issuer with a good asset and reputation φ will choose a quantityQ(φ) ∈ q, q∗(φ). To see this, observe that the issuer’s proceeds are increasing over eachsubinterval of quantity given in the definition of the demand curve, while her continuationvalue is fixed. This argument implies that q∗(φ) is a natural candidate equilibrium quantitystrategy given the inverse demand curve P . I denote the candidate equilibrium reporting

4By assuming that the opportunistic issuer plays the strategies that arise in the static signaling gamewhen she is known to be the opportunistic type, I am explicitly forcing this derivation to yield an equilibriumconsistent with Proposition 7 as issuer reputation decreases to zero.


q 1φ

P (q, φ)

q∗(φ)

1

p∗(φ)

0

Figure 3.3: An example of the demand curve (3.8) for a given φ. For q ≤ q investors believethe asset is the good type. For q < q ≤ q∗(φ), investors believe the asset is the good typewith some probability and the bad type with some probability. For q > q∗(φ) the investorsbelieve the asset is the bad type.


stragey as π∗(φ). In addition, I let the discounted loss in issuer value associated with adrop in reputation from φ to 0 be denoted L(φ) = γ(V (φ) − V (0)). Since the candidateequilibrium strategies are known at φ = 0, it is trivial to calculate

V (0) =1

1− γ(λ(q + γ(1− q)) + (1− λ)`), (3.9)

and hence deriving the value function V (φ) is equivalent to deriving the discounted lossfunction L(φ).

Now that I have assumed a particular functional form for the demand schedule P , I canfurther simplfy the maximization problem faced by the issuer. Specifically, I identify thefollowing four conditions that must hold in any equilibrium in which the quantity strategy isQ(φ) = q∗(φ), the reporting strategy is π∗(φ), and the demand schedule is given by equation(3.8),

q∗(φ)p∗(φ)− q ≥ γ(q∗(φ)− q) (3.10)

q∗(φ)p∗(φ)− ` ≥ L(φB)− (1− q∗(φ))γ` (3.11)

p∗(φ) = `+λ(1− `)

λ+ (1− λ)(1− φ)(1− π∗(φ))(3.12)

L(φ) =γ

1− γλ[q∗(φ)(p∗(φ)− γx)− (1− γ)(λq + (1− λ)`)

], (3.13)

where x = λ+(1−λ)`. Inequality (3.10) states that an issuer with a good asset must weaklyprefer the quantity strategy Q(φ) = q∗(φ) to the strategy Q(φ) = q. Similarly, inequality(3.11) states that an issuer with a bad type asset must weakly prefer the retention strategyQ(φ) = q∗(φ) and reporting strategy π∗(φ) to the strategy π(φ) = 1. Equation (3.12) followsdirectly from the fact that investors earn zero profits in expectation, in other words condition(4) of Definition 4, and Bayes rule. This equation must hold for all φ, so that to characterizean equilibrium, it is enough to find the quantity-price pair (q∗(φ), p∗(φ)). Finally, equation(3.13) follows from the maximization problem described by equations (3.6) and (3.5).

The next step in constructing a candidate equilibrium is to divide the interval φ ∈ [0, 1]into subintervals over which the inequalities (3.10) and (3.11) either bind, or are slack. Tothat end, I assume there exists φ and φ such that π∗(φ) = 1 and q∗(φ) = q for φ ≤ φ and

π∗(φ) = 0 and for φ ≥ φ. The one-shot deviation principal implies the inequality (3.11)must bind whenever 0 < π∗(φ) < 1, hence it must bind whenever φ < φ < φ. I assume there

exists φ such that inequality (3.10) binds for φ ≤ φ and q∗(φ) = 1 for φ ≥ φ. It will turnout that φ ≤ φ, so that the problem of deriving equilibrium strategies can be broken upinto the four subintervals of reputation, [0, φ], (φ, φ], (φ, φ], and (φ, 1]. Figure 3.4 shows theproposed decomposition of the interval with the corresponding constraints on the candidateequilibrium strategies π∗ and q∗.

The assumption that inequality (3.10) binds for φ < φ < φ amounts to restrictingattention to the corners of the space of incentive compatible strategies. Once inequality


0 φ φ φ 1

φ ︸︷︷︸︸︷︷︸︸︷︷︸q∗(φ) = q q < q∗(φ) < 1 q∗(φ) = 1π∗(φ) = 1 ︸︷︷︸︸︷︷︸

0 < π∗(φ) < 1 π∗(φ) = 0

Figure 3.4: The interval 0 ≤ φ ≤ 1 can be decomposed into subintervals over which thecandidate strategies are either known or the inequalities (3.10) and (3.11) bind. For φ <

φ < φ, so that q < q∗(φ) < 1, inequality (3.10) binds. For φ < φ < φ, so that 0 < π∗(φ) < 1,

inequality (3.11) binds.

(3.11) binds, one can interpret inequality (3.10) as placing a lower bound on the set ofpossible equilibrium reporting strategies. For example, suppose φB is fixed and L(φB) isknown, then the reporting strategy is a known function

π∗(φ) =φ(1− φB)

φB(1− φ). (3.14)

In other words, for each level of reputation φ ∈ (0, φB) there is a single reporting strategyπ∗(φ) for which reporting a bad asset will result in an increase of reputation to φB. Moreover,since (3.11) binds and L(φB) is known, inequality (3.10) can be rearranged to get

π(φ) ≥ 1− C 1

1− φ, (3.15)

where C is some constant which may depend on φB. Figure 3.5 plots equation (3.14) andinequality (3.15) and illustrates that by assuming inequality (3.10) binds, I am choosing theminimal admissible π∗ for each φ∗.

With the partition of the unit interval of reputation described above and the four rela-tions which must hold in equilibrium given by (3.10), (3.11), (3.12), and (3.13), the problembecomes one of solving a system of equations in a certain number of unknowns. The impor-tant caveat to this approach is that to solve the equations for equilibrium strategies at givenlevel of reputation φ, I must first know the discounted loss L(φB) at the level of reputationwhich would arise from a truthful report of a bad type asset. Since φB ≥ φ for all φ, I cansolve the problem by working downwards from φ = 1.

For φ ∈ [φ, 1], the equilibrium strategies are assumed to be q∗(φ) = 1 and π∗(φ) = 0.This means the equilibrium price p∗(φ) and associated discounted loss function L(φ) followdirectly from equations (3.12) and (3.13). For convenience, let L1(φ) denote the solution to(3.13) when q∗(φ) = 1 and π∗(φ) = 0.

For φ ∈ [φ, φ), the equilibrium retention strategy is assumed to be q∗(φ) = 1, howeverthe equilibrium price must be calculated. I assume for the time being that φB ≥ φ for all


1π(φ) = φ(1−φB)

(φB(1−φ))

π(φ)

φ

π(φ)

φB 1φ

π(φ) ≥ 1− C 11−φ

Figure 3.5: Plot of inequality (3.15) and π(φ) = φ(1−φB)(φB(1−φ))

. The downward sloping curve

represents the lower bound on π(φ) imposed by (3.15). The region above this curve isthe set of all pairs (φ, π(φ)) such that the opportunistic issuer at least weakly prefers thequantity q∗(φ) at price per unit p∗(φ) implied by the strategy π(φ) and reputation φ. Theupward sloping curve is given by the definition of π(φ). The portion of the upward slopingcurve which lies above the downward sloping curve is the set of all pairs (φ, π(φ)) such thatinequality (3.15) is satisfied and inequality (3.11) binds, and this set may contain manypoints supportable by an equilibrium. To simplify the analysis, I choose the point at whichthe two curves intersect. This point gives the lowest value of π such that both inequality(3.15) and the definition of π are satisfied.


φ ≤ φ < φ.5 Thus, for φ ∈ [φ, φ), the equilibrium price p∗(φ) of a reportedly good assetsolves the equation

p∗(φ) = L1

(((1− λ)(p∗(φ)− `)p∗(φ)− `− λ(1− `)

)1

φ

). (3.16)

Equation (3.16) follows from substituting equation (3.12) and the reputation updating func-tion into inequality (3.11) (which must bind). Again for convenience, let L2(φ) denote thesolution to equation (3.13) when p∗(φ) solves equation (3.16) and q∗(φ) = 1.

Finally, I characterize the discounted loss function L(φ), reporting strategy π∗, andquantity strategy q∗(φ) of the opportunistic type issuer for φ ≤ φ < φ. To do so, I construct

a decreasing sequence starting at φ such that each element of the sequence is the level ofreputation for which the decision to truthfully report a bad type asset would lead to anincrease in reputation to the preceding element of the sequence. Formally, let φ(n) be asequence given by

φ(0) = φ (3.17)

φB(n) = φ(n− 1) (3.18)

I can combine (3.11) and (3.10) to get

q∗(φ)p∗(φ) + γ(1− q∗(φ)x) = λ(q + γ(1− q)) + (1− λ)(`+ L(φB)). (3.19)

Equation 3.19 can be thought of as a combined incentive compatibly constraint. It states thatthe expected one period proceeds from playing the strategy (q∗(φ), π∗(φ)) is exactly equalto the expected loss from doing so. Equations (3.19) and (3.13), along with the definition ofthe sequence φ(n), imply

L(φ(n)) = βnL2(φ) (3.20)

where β = γ(1−λ)1−γλ . Now suppose that the values φ(k) are known for k ≤ n − 1. Then the

strategy pair (q∗(φ(n)), π∗(φ(n))) and the level of reputation φ(n) solve the following threeequations

q∗(φ(n))p∗(φ(n))− q = γ(q∗(φ(n))− q) (3.21)

q∗(φ(n))p∗(φ(n))− ` = βn−1L2(φ)− (1− q∗(φ(n)))γ` (3.22)

p∗(φ(n)) = `+λ(1− `)φ(n− 1)

λφ(n− 1)− (1− λ)φ(n). (3.23)

Equation (3.21) defines the price quantity pairs such that the issuer with a good type assetis indifferent between issuing the quantity q∗ at a price per unit p∗ and issuing the quantityq at a price per unit `. Equation (3.22) defines the price quantity pairs such that the issuer

5This amounts to a parameter restriction, detailed in Appendix ??. This assumption can be relaxed,although with a considerable amount of extra algebra.


with a bad type is indifferent between issuing the quantity q∗ at a price per unit p∗ resultingin a reputation of zero, and issuing the quantity one, at the price per unit `, resulting in areputation of φ(n− 1). Figure 3.6 illustrates the solution to the system of equations (3.21)-(3.23). Equations (3.21) and (3.22) both define p∗ as a function of q∗. It is straightforwardto show that these two functions satisfy a single crossing property and thus admit a uniquesolution for the quantity price pair (q∗(φ(n)), p∗(φ(n))), as demonstrated by Figure 3.6(a).Similarly, equation (3.23) defines p∗ as a function of φ. Again, it is straightforward to showthat this function is decreasing in φ. Thus by setting the right hand side of equation (3.23)equal to the solution for p∗(φ(n)) from equations (3.21) and (3.22), a unique solution forφ(n) obtains, as demonstrated by Figure 3.6(b).

The above argument shows how to characterize the candidate equilibrium on a sequenceof levels of reputation defined by (3.18) up to the solution of L(φ) for φ ≥ φ. Applying thereverse of the sequence argument to values of φ(n) < φ < φ(n−1) yields a characterization ofthe candidate equilibrium for levels of reputation not lying in the above sequence. Namely,for each φ ∈ (φ(n), φ(n − 1)) there exists a φ ∈ (φ, 1] such that an issuer with currentreputation φ will have reputation φ after choosing to report a bad type asset n times in arow.

The complete derivation of this mixed strategy equilibrium is detailed in the AppendixB. The key step is to show that the system of equations given above has a solution for eachsub-interval described above. Once this is shown, it is simple to verify that the proposedstrategies are indeed an equilibrium by using the single deviation principal for repeatedgames. I close this section with a statement of existence of a mixed strategy equilibrium inthe following proposition.

Proposition 10. Suppose Assumptions 1 and 2 hold. There exist thresholds φ, φ, φ, andan equilibrium in which the opportunistic issuer

• plays the separating equilibrium strategies for low levels of reputation: Q(φ) = q and π(φ) =1 for φ ≤ φ,

• sells a larger portion of the asset than the separating quantity and misreports a bad typeasset with positive probability for mid-range levels of reputation: q < Q(φ) < 1 for φ <

φ < φ, and 0 < π(φ) < 1 for φ < φ < φ,

• sells the entire asset and always chooses to misreport a bad type asset for high levels ofreputation: Q(φ) = 1 for φ ≥ φ, and π(φ) = 0 for φ ≥ φ.

Moreover, Q(φ) is weakly increasing φ and π(φ) is weakly decreasing in φ.

3.3.5 Analysis of the mixed strategy equilibrium

In this section, I consider implications of the existence of a mixed strategy equilibriumof the game. In particular, I discuss the following questions. What selection criteria would


Curve implied by (3.22)

γ

q∗

p∗

1q∗(φ(n))q

p∗(φ(n))q(1− γ) + γ

`+ βn−1L2(φ)

γ`

Curve implied by (3.21)

(a) Equations (3.21) and (3.22) in price quantity space. The equations both definep∗ as a function of q∗ and satisfy a single crossing property which allows for a uniquesolution (q∗(φ(n)), p∗(φ(n))).

φ(n)φ

p∗

1

0

Sol. to (3.23)

Sol. to (3.21) and (3.22)

p∗(φ(n))

λ+ (1− λ)`

φ(n− 1)

(b) The solution to (3.21) and (3.22) versus (3.23). Equation (3.23) defines p∗ asa downward sloping function of φ, thus giving a unique solution (φ(n), p∗(φ(n))).

Figure 3.6: The solution to the system of equations (3.21)-(3.22). The single crossings shownin panels (a) and (a) imply that the solution is unique.


I need to predict that a mixed strategy equilibrium of the game would obtain in the data?What do the strategy and price functions for the particular equilibrium derived above implyabout how information is revealed in the model? And finally, is this equilibrium uniqueamong mixed strategy equilibria, and if so, what are the useful comparative statics? Thesequestions lead to empirical implications, some of which I will examine in Chapter 4.

The model admits multiple equilibria. This means that in order to understand theemprical implications of the model, one most have a consistent method of selecting amongdifferent equilibria. For static signaling games, the D1 refinement of Cho and Kreps (1987)provides such a selection criterian. In my dynamic setting it is not clear how to apply theD1 refinement, so I take the following simple approach. Observe that by construction, themixed strategy equilibrium delivers at least as much per period payoff as the separatingequilibrium when reputation alone is not sufficient to implement full market discipline. Thisobservation leads directly to the following corollary of Proposition 10.

Corollary 2. Suppose Assumptions 1 and 2 hold. The mixed strategy equilibrium deliversweakly greater value to both issuer types for all levels of reputation.

I now move on to discussing the properties of the mixed strategy equilibrium. Theequilibrium quantity issued in the mixed strategy equilibrium q∗, is an increasing function ofreputation. Figure 3.7 plots q∗ versus reputation φ. For φ ≤ φ, reputation is too low to makemisreporting a bad type asset attractive to the opportunistic underwriter, so Q = q and theequilibrium reduces to the static signaling equilibrium of Proposition 6. For φ ≤ φ ≤ φ theequilibrium quantity must make the opportunistic issuer indifferent, conditional on facinga good type asset, between signaling the true type of the asset by choosing the quantityq versus issuing q∗. This implies that q∗ must be a strictly increasing function of φ forφ ≤ φ ≤ φ since the equilibrium price p∗ is a decreasing function of φ over this interval. The

equilibrium quantity issued reaches one at φ. The shape of q∗ leads directly to the followingimplication.

Implication 1. Suppose Assumptions 1 and 2 hold. Issuer reputation and retention shouldbe negatively correlated both conditionally given the issuer’s report of asset quality and un-conditionally.

Implication 1 is useful in that it provides a testable empirical prediction. If both signal-ing and reputation are available as a means of credible information transfer and the marketconditions are such that reputation alone cannot enforce, then the amount of signaling em-ployed by the issuer should be decreasing in her reputation, conditional on her public reportof asset quality. At the same time, the implication holds unconditionally in that althoughthe issuer’s retention does not depend on her reputation when she reveals that the asset isthe bad type, a negative, though less so, correlation between her reputation and retentionstill obtains.

The next interesting implication concerns the flow of information in equilibrium. In astandard static signaling model, the quantity issued is a perfect signal of asset quality. How-ever, in the equilibrium presented above, quantity is a signal of issuer reporting strategy.


Figure 3.7: A plot of q∗(φ) versus φ. For φ ≤ φ, the issuer simply issues the quantity from the

least cost separating equilibrium. For φ ≤ φ ≤ φ, the issuer chooses an increasing quantity

in her reputation. For φ > φ the issuer chooses to sell the entire asset.

This interpretation of Q means that given a level of reputation φ, the investors can inferthe reporting strategy π(φ) of the opportunistic issuer by observing issuer retention. Wheninvestors observe a relatively large Q, they can infer that the probability the issuer will mis-report a bad asset is high, and will price assets accordingly. Hence, the opportunistic issuerfaces a trade-off between choosing a perfect signal of asset quality, which implies no lemonsdiscount, and choosing an imperfect signal of asset quality, which gives the issuer the abilityto sell a larger fraction of the asset to the investors. This trade-off importantly depends onthe current level of issuer reputaion. Figure 3.8 shows the equilibrium reporting strategyof the opportunistic issuer. For φ ≤ φ, an increase in reputation increases instantaneousgains to misreporting a bad type asset, and hence the issuer increases the probability shewill indeed misreport. The issuer offsets the price impact of this increase by retaining lessof the asset. The shape of π leads to the following implication.

Implication 2. Suppose Assumptions 1 and 2 hold. An asymmetry of information shouldpersist ex post for high levels of reputation. That is, investors cannot infer the issuer’sprivate information from retention alone.

Implication 2 implies that observing asset performance after issuance provides investorswith valuable information. This is important in the context of allegations of fraudulentactivity of issuer’s. If investors could perfectly infer an issuers private information fromobserving issued quantity, there would be no scope for fraud. However, since the mixed


Figure 3.8: Plots of π(φ) versus φ. For φ ≤ φ the issuer simply plays the least cost separating

equilibrium and never misreports a bad type asset. For φ ≤ φ ≤ φ the issuer increases the

probability she will misreport a bad type asset as her reputation increases. The kink at φarises from the constraint that quantity cannot be greater than one. Finally for φ > φ theissuer always misreports a bad type asset.


Figure 3.9: A plot of p∗(φ) versus φ. For φ ≤ φ, the opportunistic issuer never reports that abad type asset is the bad type, hence the equilibrium price for a reportedly good type assetmust be one. For φ ≤ φ ≤ φ, the equilibrium price is a decreasing function of reputation.For relatively low (high) levels of reputation, the potential gain from misreporting a badtype asset is relatively small (large) since any probability that the issuer misreports a badtype asset impacts prices more (less) at low (high) levels of reputation.

strategy equilibrium implies that an ex post asymmetry of information is possible, the issuerknowingly misreports asset type and fraud is feasible.

The fact that an asymmetry of information can persist post issuance leads to an interest-ing observation about the investors beliefs on asset quality as it relates to issuer reputation.The opportunistic issuer’s equilibrium reporting strategy implies that p∗ is a U-shaped func-tion of φ. Figure 3.9 plots the equilibrium price function. For φ ≤ φ, the opportunisticissuer never reports that a bad type asset is the good type, hence the equilibrium price for areportedly good type asset must be one. For φ ≤ φ ≤ φ, the equilibrium price is a decreasingfunction of reputation. For relatively low (high) levels of reputation, the potential gain frommisreporting a bad type asset is relatively small (large) since any probability that the issuermisreports a bad type asset impacts prices more (less) at low (high) levels of reputation.Hence, in equilibrium the investors place a low probability on the issuer misreporting a badtype asset for low levels of reputation and the equilibrium price is high. For φ ≥ φ theinvestors know that an opportunistic issuer will always misreport a bad type asset, but bydefinition the probability that investors are facing an opportunistic issuer decreases as φincreases , so equilibrium price increases with φ for φ ≥ φ. The shape of p∗ leads to thefollowing implication.


Implication 3. The investors’ beliefs about the quality of the asset is u-shaped in the issuer’sreputation, conditional on the report of asset quality.

The final aspect of the mixed strategy equilibrium which requires some discussion is theupper threshold φ that represents the level of reputation past which the opportunistic issuerwill “cash in” on her reputation. For φ ≥ φ equilibrium proceeds from securitization will behigh regardless of the opportunistic issuer’s reporting strategy, so that if the opportunisticissuer achieves a reputation φ ≥ φ, she strictly prefers to report all assets as the goodtype. This outcome of the model is somewhat paradoxical when considering the benefits ofreputation effects for providing incentives for truth telling. Once an opportunistic issuer hasa high enough reputation, her optimal strategy under the mixed strategy equilibrium callsfor a complete lack of reporting discipline.

3.4 Extensions

3.4.1 Binary signal space

In this section I consider the case when the issuer may not divide the asset and hencemust sell the entire asset to investors or retain the asset. I maintain the definition equilibrium,with the additional restriction of the strategy space that Q ∈ 0, 1. Assumption 1 impliesthat when the issuer has zero reputation she will only sell bad type assets since there is nomechanism that allows the separation of good type assets from bad and the pooled priceis lower than the issuer’s value for retaining the asset. Thus, the only equilibrium of thegame when reputation reaches zero is for the issuer to retain all good assets and sell all badassets. This means that losing reputation in this setting results in a greater loss of value thanwhen the issuer can signal her private information through costly information regardless ofher reputation. As a result, the indivisibility of assets acts as a commitment mechanismallowing the loss of reputation to be a more powerful incentive mechanism.

First I consider the existence of a truth telling equilibrium in this new setting.

Proposition 11. A truth telling equilibrium exisits if and only if γ ≥ λ1−` .

Comparing Proposition 11 with Proposition 8, it is clear that a looser restriction on pa-rameters is required to implement truth telling when the issuer’s quantity choice is restrictedto all or nothing. This is precisely because reputation is a more powerful incentive mecha-nism in this new setting and hence the issuer can credibly commit to truthfully reportingasset type for a wider range of parameters.

It remains to characterize an equilibrium for parametrizations which do not allow theexistence of a truth telling equilibrium, i.e. when γ < λ

1−` . The method for constructing isessentially the same as in Section 3.3.4. First, I assume a particular form for the demandschedule

P (φ, q) =

1 for q = 0p∗(φ) for q = 1.

(3.24)


Then the analogs to equations (3.10), (3.11), and (3.13) in this setting are

p∗(φ) ≥ γ (3.25)

p∗(φ) ≥ `+ γ(V (φB)− V (0)) (3.26)

L(φ) =γ

1− γλ(p∗(φ)− (1− γ)V (0)) (3.27)

for Q = 1. Inequality (3.25) states that the issuer must prefer to sell a good type assetat price p∗(φ) rather than retain it. Inequality (3.26) states that the issuer must prefer tomisreport a bad type asset and sell it at price p∗(φ) rather than accurately report it andsell it at price `. Equation (3.27) simply follows from equations (3.5) and (3.6). Inequalities(3.25) and (3.26) only apply to levels of reputation for which there is a positive probabilitythe issuer misreports a bad type asset and still chooses to sell a good type asset. I assumethere exists a subinterval for which (3.26) binds, so that the issuer misreports the bad typeasset with probability strictly between zero and one. Then finding equilibrium strategiesreduces to a problem of solving a system of equations in a certain number of unknowns. Thefollowing proposition summarizes the resulting mixed strategy equilibrium.

Proposition 12. Suppose 1 − ` > γλ. There exists φ′ and φ′ and an equilibrium in whichthe opportunistic issuer

• retains good assets and sells bad assets for low levels of reputation, that is

Q(φ) = 0 for φ ≤ φ′,

• sells both asset types and misreports a bad type asset with positive probability for mid-range levels of reputation, that is

Q(φ) = 1, and 0 < π(φ) < 1 for φ′ < φ < φ′,

• sells both asset types and always misreports a bad type asset for high levels of reputation,that is

Q(φ) = 1 and π(φ) = 0 for φ ≥ φ′.

Moreover, π(φ) is weakly decreasing.

Restricting of quantities to the set 0, 1 substantially changes the equilibrium reportingstrategy relative to the general case. Figure 3.10 compares the equilibrium reporting strate-gies for Propositions 10 and 12. The first difference is that there is a discontinuity in thereporting strategy for the indivisible asset case. This discontinuity exists because quantitiescannot continuously adjust with reputation to make the issuer indifferent between retaininga good asset and selling it in its entirety. Notably, the indivisibility of assets causes the issuerto refrain from sending inaccurate reports at levels of reputation for which she would do soif she could retain part of the asset. In other words, φ < φ′. Similarly, the issuer will send


Figure 3.10: A comparison of π(φ) for the divisible and indivisible asset settings. The solidcurve is the reporting strategy for the divisible setting, while the dashed curve is the reportingstrategy for the indivisible setting. φ < φ′ implies that the indivisibility of assets causes theissuer to refrain from sending inaccurate reports at levels of reputation for which she woulddo so if she could retain part of the asset. Similarly, φ < φ′ implies the issuer will send aninaccurate report with probability 1 for lower levels of reputation when she can retain a partof the asset than when the asset is indivisible.


an inaccurate report with probability 1 for lower levels of reputation when she can retain apart of the asset than when the asset is indivisible. In other words, φ < φ′.

The differences between reporting strategies described above have important implica-tions for policies which aim to reduce the probability that the issuer sends an inaccuratereport. Disallowing issuer retention is actually one possible mechanism to decrease theamount of lying in equilibrium. The drawback to such a restriction is that issuer payoffswill decrease. In addition, for low levels of reputation, the market for ABS will be a marketfor lemons.6 Thus the issuer, conditional on having a low initial reputation, receives a lowervalue for the game when the asset is indivisible than when the asset is perfectly divisible.However, this means that the cost of a loss in reputation is potentially greater when theissuer is not free to choose quantity.

If the opportunistic issuer starts with a high enough reputation, she will send an inac-curate report at least part of the time. In this case the relationship between the probabilityof sending an inaccurate report and the divisibility of assets is not clear. For intermediatelevels of reputation the issuer will want to sell only part of the asset to investors but is con-strained to sell the entire asset, and thus may lie with greater frequency. However for highlevels of reputation, the threat of losing reputation becomes more powerful for the indivisibleasset case because the market will revert to a market for lemons when the issuer loses herreputation. As a result, the issuer has less of an incentive to send an inaccurate report forthe indivisible asset case when she has a high reputation.

3.4.2 Risky assets

In this section I consider a richer specification for the cash flow of assets. As above, Iassume there are two types of assets: good and bad. Both asset types produce a cash flow inthe next period contained in the set xl, xh with xl < xh. Good assets have a probabilitypG of producing a cash flow xh such that

pGxh + (1− pG)xl = 1

while bad assets have a probability pB of producing xh such that

pBxh + (1− pB)xl = `.

These asset cash flows mean that the expected cash flow of the asset conditional on its typeremains unchanged.

The next step is to characterize the reputation updating function in this new setting.Lemma 1 still applies since its proof does not depend on the distribution of asset cash flows.

6Downing, Jaffee, and Wallace (2009) show that the “to be announced” MBS market is indeed a marketfor lemons. This finding follows from the indivisibility of assets and the anonymity of issuers. The implicationof the current model is that such a finding can persist in a setting with reputation effects. This does notdepend on the stylized type of reputation I consider here and would obtain even when punishment strategiesare allowed since the parameters are such that punishment strategies are not sufficient to support a truthtelling equilibrium.


So once again, reputation is not updated during the securitization phase and her currentquantity choice does not affect her current reputation. The equilibrium reputation updatingfunction is thus given by Bayes rule as follows

f(g, xh, φ) = φS =φ

1 + (1− φ)(1− π(φ))1−λλ

pBpG

(3.28)

f(g, xl, φ) = φF =φ

1 + (1− φ)(1− π(φ))1−λλ

1−pB1−pG

if pG < 1 and 0 otherwise (3.29)

f(b, x, φ) = φB =φ

φ+ (1− φ)π(φ)(3.30)

The first observation is that the separating equilibrium of Proposition 9 still holds. Inthat equilibrium, the assets ex post cash flows do not affect the future play of the game.This is precisely because the issuer credibly reveals her private information about the asset’sexpected cash flow via her quantity choice. Therefore, no asymmetry of information aboutthe current asset persists once the current period quantity choice is observed. As a result,earlier statements about the inability of the separating equilibrium of Proposition 9 to explainthe existence of fraud do not depend on the stylized asset cash flow distribution of theprevious section.

Since the assets expected cash flows have not changed, the existence of a pooling equi-librium remain unchanged as well. Specifically, Assumption 1 still implies that poolingequilibria do not exist. That assumption states that the issuer would rather retain a goodtype asset then sell it at a price equal to the ex ante expected cash flow of the asset.

The next task is to characterize the conditions under which a truth telling equilibriumexists for the game with risky asset cash flows. First consider the case when pG = 1. This willimply that investors will know that the issuer is the opportunistic type when a cash flow ofxl follows a report that the asset is the good type. In this case, the restriction on parametersneeded for the existence of a truth telling equilibrium again places an upper bound on thediscount rate of the issuer, however now the bound also depends on the probability that thebad type asset produces the cash flow xh as is stated in the following proposition.

Proposition 13. There exists a truth telling equilibrium if and only if γ ≥ 1λ(1−pB)+`

An important feature of Proposition 13 is that the restriction required for the existenceof a truth telling equilibrium is more strict for the case of risky asset cash flows than forthe setting of the previous section. This is because there is still a chance an opportunisticagent may not get “caught” misreporting the bad type asset. Thus, the issuer has lessof an incentive to accurately report a bad type asset when asset cash flows are risky. Inreality asset cash flows are indeed risky, making the empirical observability of truth tellingequilibrium less likely.

If pG < 1, then a cash flow of xl does not reveal an inaccurate report. This againdecreases the issuer’s incentive to accurately report the bad type asset since even if the asset


yields the cash flow xl, the investors cannot be sure that the issuer is indeed the bad type.This leads directly to the following negative result.

Proposition 14. If pG < 1, a truth telling equilibrium does not exist .

The intuition behind Proposition 14 is as follows. Suppose that a truth telling equi-librium did indeed exist and pG < 1. Then the equilibrium reputation updating functiongiven in equations (3.28) - (3.30) imply that φS = φF = φB = φ. Thus, a cash flow of xl

does not affect reputation and the issuer would never lose any reputation from misreportingbad type assets. At the same time, if investors believe that the issuer is following the truthtelling strategy, then they must price a reportedly good type asset accordingly, leading toone period gains for the issuer from misreporting a bad type.

The non-existence of a truth telling equilibrium for the case of risky asset cash flowssuggests that other equilbria, such as the mixed strategy equilibrium of the previous section,may exist. Unfortunately, this setting will not admit a solution technique for a mixedstrategy equilibrium like the one presented in the previous section. To see this, assume thatthe demand schedule is given by equation (3.8). Then (q∗(φ), π∗(φ)) is a natural candidateequilibrium strategy pair. However, the analog to equation (3.11) is

q∗(φ)p∗(φ)− ` ≥ γ(pB(V (φB)− V (φS)) + (1− pB)(V (φB)− V (φF )))− (1− q∗(φ))γ`(3.31)

which depends on the value function evaluated at three different levels of reputation. Hencemore constraints on equilibrium strategies would be needed to pin down a particular mixedstrategy equilibrium.

3.4.3 Path dependent beliefs

In this section, I consider investor beliefs that may depend on the entire history ofthe game, rather than just current reputation and current quantity. The investors’ demandcurve at time t is then given by a function Pt(q,Ht) : [0, 1]×Ht → [0, 1] where Ht is the setof all possible histories of the game up to time t. The issuer then implements a reportingstrategy πt(Ht) : Ht → [0, 1] and a quantity strategy Qt(Ht) : Ht → [0, 1]. I maintain theassumptions that the issuer always reports a good type asset as good, and always issues thequantity one when reporting the asset is the bad type.7 The definition of equilibrium in thissetting replaces conditions (4) and (5) of Definition 4 with

5. Investors earn zero expected profits: Pt(Qi(Ht),Ht) = E[Xt+1|Ht, Q

i(Ht)] for i ∈H,O

6. An equilibrium is separating if Pt(Qi(H),Ht) = 1.

7These assumptions are without loss of generality as a variant of Lemma 1 holds in this more generalsetting.


The first result is that an equilibrium satisfying Definition 4 will satisfy the definitionof equilibrium in this more general setting.

Proposition 15. Suppose the quadruple (P , QH , π, QO) satisfies Definition 4, then there ex-ists an equilibrium of the game with history dependent strategies such that P (q,Ht) = P (q, φ),QHt (Ht) = QH(φ), πt(Ht) = π(φ), and QO

t (Ht) = QO(φ) where φ = P(Issuer is Honest Type|Ht)

Proof. Observe that if strategies are defined as in the proposition and

φ = P(Issuer is Honest Type|Ht),

then conditions (4)-(5) of Definition 4 imply conditions (4)-(5) of this section.

The above proposition is not surprising. It simply states that stationary Markov equilib-ria are also equilbria under the more general definition. It is of greater interest to determinewhether there are equilibria under the more general definition that do not obtain under Def-inition 4. I show that there are truth telling equilibria under the general definition that donot exist under Definition 4, however the parameter restriction required for their existencecoincides exactly with that of Proposition 8.

Proposition 16. Suppose φ0 = 0. Then there exists a truth telling equilibrium with Qt(Ht) >q for some Ht if and only if γ ≥ 1

λ+`.

Proof. Note that since φt = 0 for all t ≥ 0 by Bayes rule, hence it is enough to considerthe strategies of the opportunistic type issuer. First suppose there exists a truth tellingequilibrium given by Qt(Ht) and πt = 1. Let Q be defined as follows

Q = supQt(Ht)|t ≥ 0 and Ht ∈ Ht. (3.32)

Note that Q > q. For all ε > 0, there exist (t,Ht) such that Q−Qt(Ht) < ε by the definitionof supremum. Let (t,Ht) be such a pair. The one-shot deviation principal implies that

Qt(Ht) + (1−Qt(Ht))γ`− ` ≤ Lt (3.33)

where Lt is the discounted loss faced by the issuer after misreporting a bad type asset giventhe history Ht. By the definition of Q, I have

Lt ≤ γ

(λ(Q+ (1− Q))γ + (1− λ)`

1− γ− λ(q + (1− qγ) + (1− λ)`

1− γ

)= γλ(Q− q). (3.34)

ButQt(Ht) + (1−Qt(Ht))γ`− ` = (1− γ`)(Q− ε− q). (3.35)

This implies that for all ε > 0

(1− γ`)(Q− ε− q) ≤ γλ(Q− q) (3.36)


which implies γ ≥ 1λ+`

since Q > q.

Now suppose γ ≥ 1λ+`

. I’ll show that a truth telling equilibrium is given by Qt = 1,πt = 1, P (q,Ht) = 1 if no misreports have been made and

P (q,Ht) =

1 if q ≤ q` if q > q

otherwise. Note that the continuation value of the issuer just depends on whether or notshe has made a misreport. Let L be the discounted loss in continuation value faced by theissuer if she misreports a bad type asset then

L = γ

(λ+ (1− λ)`

1− γ− λ(q + (1− q)γ + (1− λ)`

1− γ

)= γλ(1− q). (3.37)

To see that the above strategies indeed constitute an equilibrium observe that

L = γλ(1− q) = γλ1− `

1− γ`≥ 1− `.

Thus the discounted loss in continuation value is greater than or equal to the one-shotgains from misreporting a bad type asset and the above strategies constitute a truth tellingequilibrium for φ0 = 0.

Proposition 16 shows that although allowing for path dependent strategies does meanthat a truth telling equilibrium is supportable without the type of reputation considered inthis paper, the restriction on the parameters required for truth telling does not change.

3.5 Conclusion

In this chapter, I have presented a model of an issuer of a generic asset which unifiessignaling and reputation effects. In my model, a lemons problem arises due to asymmetricinformation about the quality of assets. Partial retention of an asset by the issuer is a crediblesignal of asset quality since the issuer is impatient relative to investors causing retention tobe costly. Imperfect information over issuer preferences induces a market reputation for theissuer. A high reputation can increase payoffs for the issuer by reducing the fraction of theasset the issuer retains in equilibrium while decreasing the lemons discount relative to anidentically structured sale by an issuer with a low reputation. Reputation effects do notimply that an issuer will be more likely to perfectly reveal her information.

The implications of my model call into question the benefits of reputation as a sub-stitute for regulation and oversight for imposing market discipline. Although conceptuallyappealing, the assertion that issuers of ABS, for example, will behave in the best interestsof the wider markets as a consequence of protecting their reputations misses an important


point. The benefit of having a good reputation may be due to the ability to “cash in” on ahigh reputation in the future. In the case of my model, an opportunistic issuer will cash in ona good reputation by misreporting bad type assets. In contrast, signaling in the absence ofreputation can force issuers to reveal the true full information value of any assets underlyingABS at the cost of reducing equilibrium payoffs.

While I mostly discuss the model by referring to asset backed securities, the theorydeveloped in this paper applies equally well to other important financing problems. The keyfeatures of the model are that the issuer has valuable private information, a means by whichto signal that information in a single period, and a means by which to gain a reputation foraccurate reports. For example, the model could refer to a venture capitalist raising fundsfrom limited partners. She may need to maintain a larger stake as general partner if she doesnot yet have a good track record of matching investment projects with stated fund goals.Similarly, a private equity firm may need to put up a larger amount of capital to implementa leveraged take over if that firm does not have a long history of accurate analysis of targetfirm prospects. Finally, both a private equity firm and venture capitalist may at some pointfind it advantageous to exploit a good reputation for one period gains.

I do not make specific policy suggestions since the model is silent on the effects of differ-ent equilibrium strategies on markets for underlying assets and collateral, like the primarymortgage market. However, depending on the goals of a regulator, the model offers thefollowing advice. If the policy goal is to ensure accurate information disclosure, then a reg-ulator should adopt a policy which encourages perfect signaling of asset quality. Providinga legal means for issuers to publicly disclose their holdings and commit to hold them tomaturity could increase accurate information transmission. In contrast, if the policy goal ismaximizing payoffs for issuers then reputation and signaling should both be facilitated, i.e.through a centralized repository of past deal performance and a credible means to revealissuer holdings.

63

Chapter 4

CMBS issuer reputation and retention

4.1 Introduction

The theoretical results presented in the previous chapter lead to important empiricalpredictions that can and should be tested. In this chapter I will examine Implication 1 of theprevious chapter, namely are issuer reputation and retention negatively correlated. This taskbrings up two main challenges. First, I must choose a particular market in which to makethis inquiry, and as the title of this chapter suggests this will be the commercial mortgagebacked securities (CMBS) market. Second, I must define consistent ways of measuring bothissuer reputation and retention. The main result of this exercise is that issuer reputationand retention are indeed negatively correlated, controlling for other issuer characteristics,deal characteristics, and time trends.

The (CMBS) market represents a unique laboratory to study the relationship betweenreputation and signaling via retention for two main reasons. First, it is concentrated amonga small number of issuers so that a given issuer has the opportunity to develop a reputation.Second, unexpectedly poor performance in this market occurred before the most recentfinancial crisis, leading to potential updates to some issuers’ reputation among investors.

In order to conduct the analysis I need measurements of both issuer reputation andretention. To measure issuer reputation, I track the number of downgrades an issuer has hadon past deals. This measure is motivated by the model: issuer reputation should be updatedaccording to past performance. This differs from previous empirical work, for example Linand Paravisini (2010) or Titman and Tsyplakov (2010) TS, which use market based proxiesfor reputation like market share or stock price performance. To measure retention I usea particular feature of the CMBS market. CMBS deals often contain an unrated tranchewhich is likely retained by the issuer for some time. Rentention is then the proportion of thebelow investment grade principal which is unrated. To my knowledge, there has not beensignificant empirical attempts to test signal based thoeries of security design using an actualmeasure of issuer retention (signaling). Thus, my measure of issuer retention is a potentiallyunique and valuable tool for emprical work going forward.

Chapter 4. CMBS issuer reputation and retention 64

One further difficulty is that Implication 1 is conditional on the issuers report of assetquality, which is difficult to observe for the econometrician. However, it will be more difficultto detect this effect in the data without conditioning on the report of asset quality. Hence, ifan unconditional analysis suggests that the effect is present, the result should only be madestronger by including information about the issuers report of asset quality.

This negative correlation between issuer reputation and retention documented in thischapter is similar in spirit to effects discovered by Lin and Paravisini (2010) and Sufi (2007) inthe syndicated loan market. Lin and Paravisini show that an exogenous shock to reputation,namely exposure to the accounting fraud scandals of the early 2000’s, causes an increasein lead arranger share of a syndicated loan. Theyargue that this means that a change inreputation can affect the structure of incentive contracts used to mitigate moral hazardproblems induced by the unobservable costly monitoring technology of the lead arranger.In the current setting, monitoring is not a primary concern due to the specific nature ofABS in which managers have no control rights once a security is issued, and as a resultthe motivating friction is adverse selection rather than moral hazard. Sufi finds that leadarrrange capital contribution decreases in market share, although never reaches zero. Thisresult is likely due to adverse selection problems decreasing with reputation (market share)while some capital contribution is always required to combat moral hazard.

Another related article is Titman and Tsyplakov (2010), who investigate the link be-tween mortgage originator performance, commercial mortgage spreads and default rates, andCMBS structure. They find originator’s with poor stock price performance originate lowerquality mortgages which also end up in CMBS deals with more subordination (less AAArated principal). While the focus of their paper is on mortgage originators, the focus ofthe empirical work in this paper is on CMBS issuers. Moreover, I find that by measuringissuer reputation in terms of real past performance, rather than in terms of a market basedproxy, there is no affect of issuer reputation on the size of the AAA tranche. To the extentthat a CMBS issuer is often the same entity as a mortgage originator, this result indicatesthat Titman and Tsyplakov’s market based measure of originator reputation is substantivelydifferent than my performance based measure

4.2 Instituational Background

In the CMBS market, commercial mortgages are pooled together and placed in a passivetrust by a sponsoring entity, usually an investment bank. Bonds of differing seniority andmaturities, often called tranches or classes, are then sold by the sponsor to various investors.I will refer to a series of tranches all backed by the same mortgage pool as a deal. Thestructure of a typical CMBS deal is similar to the perhaps more familiar residential mortgagebacked security (RMBS), in that pre-specified rules govern the distribution of principal andinterest payments to the classes.

Before the bonds are marketed to investors, but after the characteristics of the poolof mortgages are fixed, a rating agency will assign a rating to at least some of the classes.


Ratings are assigned given the hard information provided by the sponsor, for example mort-gage characteristics such as loan-to-value ratio, weighted average coupon, and geographiclocation, but could also take into account any soft information the sponsor chooses to reveal.The ratings agency might indicate to the sponsor that certain classes must be given morecredit support, i.e. have a larger amount of principal with lower seniority in the deal, inorder to obtain a certain rating. Once ratings have been assigned, the sponsor creates aprospectus that contains both hard and soft information as well as the ratings assigned toeach class. This prospectus is a available to any investor interested in purchasing a classfrom a given deal.

The sponsoring entity of a particular CMBS deal plays the role of the issuer in mymodel. Although some sponsors originate their own mortgages, other sponsors simply pur-chase mortgages from originators. In any case, sponsors will typically hold an inventory ofmortgages waiting to be securitized. At this point, even sponsors who have played no rolein the loan origination process have the opportunity to gain valuable, and potentially non-verifiable, information about the probability that a given mortgage will default and whatrecovery rates might be given default.

Although CMBS have existed since the mid 1980s, the market truly began to expandin 1995. The yearly issuance of private label (deals not issued by one of the governmentsponsored entities) expanded from around $15 billion in 1995 to more than $230 billion in2007.1 With this expansion, the CMBS market became a major source of capital for thecommercial real estate market.

4.3 Data and Measurement

The data come from the Lewtan Technologies ABSnet dataset, Bloomberg, and Com-merical Mortgage Alert and consist of structure and performance data for 703 CMBS dealsissued between 1989 and 2008. Descriptive statistics for the dataset are display in Table4.1. Two important features of the data to note are the mean deal size of $1.43 billion andthe mean weighted average loan to value (WALTV) of 66%. The mean deal size indicatesthat the average CMBS deal is quite large and that the data comprise around one trilliondollars of total issuance. Thus, although the CMBS market is smaller than the prime andsubprime RMBS markets, it is still quite large. The mean WALTV indicates that commercialmortgages tend to be less leveraged than subprime residential mortgages. Although an exactcharacterization of the overall size of the CMBS market is not readily available, comparingthe aggregate issuance contained in these data with other sources2 indicate that the datarepresent a nearly exhaustive sample. My analysis can only be run on a subsample of 457deals due to missing data for some covariates. Characteristics of this subsample match thoseof the total sample well, implying my subsample is representative of the larger sample.

1Source: www.CMalert.com2For instance, www.CMalert.com, or www.CMBS.com.


Table 4.1: Summary statistics for pool and deal characteristics. Percentage AAA is theproportion of principal rated AAA at issuance. Deal size is the total face value of all tranchesat issuance in $ billions. Weighted Average Coupon is the weighted average coupon rate ofthe mortgages in the pool. Weighted Average LTV is the weighted average loan-to-valueratio in the pool. Debt Service Ratio is the weighted average ratio of debt service paymentsto net operating income. Percentage office is the percentage of mortgages collateralized byoffice space. Conduit Dummy indicates deals comprised of conduit loans. # Loans is thenumber of loans in the pool (in hundreds). # Deals is the number of deals issued by theissuer prior to the current observation deal.

Variable N Mean Median StandardDeviation

Percent AAA 696 0.765 0.827 0.199Percent of Below Investment Grade Unrated 703 0.346 0.313 0.278

Total Issuer Downgrades 703 0.551 0.000 1.132Issuer time in Sample 703 5.147 4.528 4.016

Weighted Average Loan Spread to Treasury 659 0.019 0.018 0.011Weighted Average Loan-to-Value 636 0.635 0.683 0.154

Debt Service Coverage Ratio 590 1.639 1.490 0.561Percent Office 595 0.293 0.266 0.180Percent Hotel 521 0.108 0.078 0.146

Percent Industrial 547 0.087 0.073 0.069Number of Loans in Deal (in 100s) 645 2.657 2.090 2.059

Average Loan Size in Deal (in $Mils) 645 31.670 7.310 125.883Dummy for Conduit Deal 645 0.766 . .


Table 4.2 provides an example of a deal structure, using the deal “Morgan StanleyCapital Inc. 2003-TOP11”. I will discuss signal score shortly. This deal structure is repre-sentative of the deals in the dataset and consists of 21 tranches, with 12 tranches making up97% of the securitized principal rated above investment grade at issuance. Tranches receiveprincipal payments in alphabetical order such that each tranche receives its scheduled princi-pal payment before any junior tranches receive principal. Tranches X-1 and X-2 are “interestonly” and receive interest based on a schedule of notional values detailed in the prospectuswith an initial notional equal to the initial balance listed in the table. Tranches R-I throughR-III are reserve tranches and only receive principal in the event that all other tranches areretired. The tranches rated below investment grade are the most informationally sensitiveportion of the deal and can be thought of as the “asset” in the model.

The primary implication of the model is that issuers can substitute between costlysignaling and reputation as means to increase the price of their marketed securities. Twoprincipal empirical challenges emerge from this implication. First, although market partic-ipants may observe the identity of the eventual owner of each tranche, I do not. As suchit is difficult to measure the fraction of the asset the issuer retains. Second, there is noestablished measure of the market reputation of a given issuer.

To deal with the first challenge, I appeal to the following stylized conditions of theCMBS market. Many deals contain a certain tranche (or number of tranches) which donot receive a rating. These tranches are referred to as “stipulations” or “stips” and areusually retained by the issuer. To measure the percentage of retention in manner which iscomparable across deals, I divide the principal balance of the unrated tranches by the totalprincipal balance of all tranches rated below investment grade and call this variable SignalScore. In this way I am measuring the fraction of the informationally sensitive portion ofthe deal, or the asset of the model, retained by the issuer. To the extent that Signal Scoredoes not accurately measure issuer retention, it is not clear that the error in measurementshould be correlated with an issuers reputation. Thus measurement error for issuer retentionshould bias against finding a result.

During the sample period, the CMBS market was going through many structural changeswith respect to the average levels of subordination within deals. Figure 4.1(a) show theyearly average subordination levels for 1996 through 2008. Of particular note is the largeincrease in the percentage of AAA during that period, from around 56% in 1993 to around80% in 2008, with a similar trend in the percentage rated above investment grade.3 Sincemany deal balances exceed $1 billion, this represents a large dollar-amount increase in AAACMBS tranche issuance. In addition, the average signal score increased over this period, asdemonstrated by Figure 4.1(b). This means that the increase in AAA and above investmentgrade was at the expense of the tranches rated just below investment grade. These trendssuggest that CMBS market characteristics varied substantially over time. To account for thisapparently systematic time series variation, I will include either a time trend or quarterly

3This increase could be due to an improvement in underlying securitized collateral or a liberalization ofrating standards. For evidence supporting the latter explanation see Stanton and Wallace (2010).


Table 4.2: The structure of the deal Morgan Stanley Capital Inc. 2003-TOP11. This dealconsists of 21 tranches, with 12 tranches making up 97% of the securitized principal ratedabove investment grade at issuance. Tranches receive principal payments in alphabeticalorder such that each tranche receives its scheduled principal payment before any juniortranches receive principal. Tranches X-1 and X-2 are “interest only” and receive interestbased on a schedule of notional values detailed in the prospectus with an initial notionalequal to the initial balance listed in the table. Tranches R-I through R-III are reservetranches and only receive principal in the event that all other tranches are retired. Totals donot include interest only tranches, as those balances are only nominal and the tranches donot receive principal payments from the mortgage pool. Signal Score is the ratio of unrated(NR) principal to total principal rated below investment grade

Tranche Initial Rating Initial Balance SubordinationName (S&P) ($ Mil.) Pct.

Inv.

Gra

de

A-1 AAA 150.00 12.0%A-2 AAA 175.00 12.0%A-3 AAA 165.11 12.0%A-4 AAA 561.38 12.0%X-1 AAA 1194.88 12.0%X-2 AAA 1099.30 12.0%B AA 31.37 9.4%C A 32.86 6.6%D A- 13.44 5.5%E BBB+ 14.94 4.3%F BBB 7.47 3.6%G BBB- 7.47 3.0%

Bel

owIn

v.

Gra

de

H BB- 11.95 2.0%J B+ 2.99 1.8%K B 2.99 1.5%L B- 2.99 1.3%M CCC+ 2.99 1.0%N NR 11.95 0%R-I NR 0R-II NR 0R-III NR 0

Total: 1194.88Signal Score: 36.3%


Figure 4.1: The yearly sample average subordination levels and signals scores for 1996-2008.

1996 1999 2002 2005

CMBS Subordination Levels

Year

Per

cent

025

5075

100%

% AAA% >BBB− and <AA% < BBB−

(a) CMBS Subordination Levels at issuance over time. Note how the per-centage of princpal rated AAA is increasing over the sample period, fromabout 60% in 1996 to nearly 80% in 2008, while at the same time the per-centage of principal rated between AAA and BBB- stays somewhat constant.This indicates that either the perceived credit quality of the underlying mort-gages was increasing, or rating standards were decreasing.

Signal Score

Year

Avg

Sig

nal S

core

3040

50%

1996 1999 2002 2005 2007

(b) Average Signal Score over time. The signal score is increas-ing over the sample period. This indicates that at the same timethat the proportion of principal rated below invesment gradewas decreasing, the ratings within that portion of the capitalstructure were shifting downwards. So not only was there lesscredit support for the above investment grade classes, the creditsupport was also rated lower.


fixed effects in all regressions attempting to explain signal score.The next empirical challenge consists of measuring market reputation. This problem can

be thought of in two parts. First, I must obtain some measure of initial market reputation.Second, I must obtain some measure of changes in reputation. Since differences in initialissuer only varies cross-sectionally, I can control for initial issuer reputation by includingissuer fixed effects. To address the second part, I measure the degree to which a given issuerhas had bonds from past deals downgraded. The reasoning for equating a downgrade in apast deal to a loss in reputation is as follows. If a given issuer has had many downgrades onpast deals, then the assets underlying those deals performed worse than expected, and themarket will decrease the reputation of that issuer. Specifically I create a map from lettergrade ratings to the [0, 1] interval and measure a downgrade as the difference between theprior numeric rating and the newly assigned numeric rating of a bond. For example, if abond is downgraded from AAA to NR, my measure for the downgrade would be 1. For eachdeal, I then sum the downgrades attributable to the issuer that occurred prior to the closingdate of the deal.

4.4 Results

I now estimate the relationship between reputation and signaling using the proxiesconstructed above. Formally, I estimate the tobit regression

Sij = β0 + β1qt + β2Downgradesjt + θXi + αj + εi (4.1)

where Si is signal score of deal i, qt is either the time in years since the start of the sampleor a time fixed effect, Downgradesjt is the prior downgrades attributed to the issuer of thedeal, αj is an issuer level fixed effect and and X is a vector of controls for the characteristicsof the underlying mortgage pool and other deal level characteristics. Specifically, I includedays since the start of the sample, days since the issuer entered the sample, pool weightedaverage spread to treasury, pool weighted average loan-to-value, weighted average debt ser-vice coverage ratio, the percentage of mortgages collateralized by office, hotel and industrialproperties, the number of loans in the pool, the average loan size in the pool, a dummyfor deals comprised of conduit loans,4, and controls for the geographic composition of themortgage pool. See Table 4.1 gives summary statistics for the vector X of controls. SinceS is confined to the interval [0, 1], I estimate the coefficients of equation (4.1) using a tobitregression.

Models (1)-(5) of Table 4.3 show the results of the estimation of (4.1) for various vectorsof controls. The main coefficient of interest is that on Total Issuer Downgrades, which ispositive and significant for all specifications. This means that an increase in bad performance,i.e. downgrades, on past deals is associated with an increase in the amount of costly signalingemployed by the issuer. The dynamic model of this paper is consistent with this fact. In

4Conduit loans are commercial mortgages originated to be securitized.


contrast, the previous literature on security design in a costly signaling framework is silent asto the effect of past performance on current issues, except through an exogenously specifiedchange in beliefs about the distribution of the quality of underlying assets.

The estimates of the coefficients on the variables measuring observable pool character-istics are also of some interest. The variables with significant coefficients, other than TotalDowngrades by Issuer, are WALTV, Debt Service Coverage Ratio, and property type. Thelarge positive WALTV and the corresponding negative coefficient on WALTV squared sug-gest that the effect of observable credit quality on the signal score is highly non-linear. Forvery low levels of WALTV less signaling is employed. This effect is likely due to strongfundamental measures of credit quality being associated with less uncertainty about poolquality. In other words, the lemons problem is smaller for pools with very strong fundamen-tals since it is very unlikely that a low quality pool would attain very high fundamentals.For for mid-range values of WALTV, the pool should be of relatively high credit quality,this corresponds to the case in the model when the issuer makes the public report that themortgages are of high quality and hence more retention is employed. For very high values ofWALTV, less signaling is again being employed. This corresponds to the case in the modelin which the issuer reveals that the mortgages are of low quality and no signaling is neces-sary. The negative coefficient on Debt Service Coverage Ratio indicates that deals of higherobservable credit quality require less retention.

The property type variables could proxy for the amount of diversification across loansin the pool. A pool with a high degree of concentration in a certain characteristic is moresensitive to the issuers private information, ceteris paribus. For example, if a pool is predom-inately made up of mortgages on office space, then the performance of that pool is highlysensitive to the issuer’s private information about the office space market. As a result, theissuer will need to retain more of a such a deal to signal its quality.

The results of Table 4.3 suggest that subordination levels within the lowest rated portionof a deal are related to issuer reputation. This affect could be due to a change in investorbeliefs about the distribution of the underlying assets available to a given issuer rather thanto the substitutability of reputation and signaling. To address this concern, I replicate theregressions of Table 4.3 with the percentage of the deal rated AAA as the dependent variable.Table 4.4 shows the results for this analysis. The coefficient on Total Issuer Downgrades isno longer significant, indicating that my measure of issuer reputation is not related to theover-all level of subordination within a deal. Consequently, the results of Table 4.4 are notlikely driven by a change in investor beliefs about the quality of available assets.

4.5 Conclusion

Taken as a whole, the results of this chapter suggest there is a strong link between anissuer’s past performance and current deal structure. Specifically, an issuer of CMBS cantradeoff between using her reputation and retaining a portion of an issue to increase the priceof her offerings. This result indicates that empirical work that attempts to either explain


Table 4.3: The empirical relationship between signaling and reputation. The table displaysresults of a tobit estimation of (4.1) with various controls. The dependent variable is thepercentage of below investment grade principal which is not rated. The main coefficient ofinterest is that on Total Issuer Downgrades, which is positive and significant for all speci-fications. This result indicates that an increase in downgrades on past deals is associatedwith an increase in the amount of costly signaling, consistent with the main prediction ofthe model. Standard errors in parenthesis are clustered at the Issuer level. All models haveχ2 statistics that imply significance at the .001 level.

Percentage below investment grade not rated

(1) (2) (3) (4) (5)

Constant 0.346∗∗∗ 2.104 0.810 1.532∗∗ -1.790(0.000631) (1.775) (1.878) (0.608) (1.250)

Days Since Start of Sample -0.134 -0.154(0.123) (0.126)

Total Issuer Downgrades 0.0634∗∗∗ 0.0469∗∗∗ 0.0283∗∗∗ 0.0218∗∗∗ 0.0212∗∗

(0.00641) (0.0117) (0.00734) (0.00817) (0.0106)Issuer Days in Sample 0.140 0.170 -0.152 0.135

(0.124) (0.128) (0.111) (0.131)Weighted Average Spread -1.425 -1.040 -1.703 -2.099

(2.479) (2.076) (5.411) (4.059)Weighted Average Spread Squared -5.034 -2.527 -11.17 18.42

(34.23) (23.03) (53.95) (39.39)Weighted Average LTV 0.693 5.858∗∗ 0.463 5.850∗∗

(0.823) (2.887) (0.665) (2.838)Weighted Average LTV Squared -0.584 -4.904∗∗ -0.404 -4.960∗∗

(0.898) (2.224) (0.788) (2.144)Debt Service Coverage Ratio -0.104 -0.115∗∗∗ -0.0994 -0.109∗∗

(0.0834) (0.0433) (0.0828) (0.0510)Percentage Office 0.194 0.292∗∗

(0.131) (0.121)Percentage Hotel 0.131 0.0897

(0.176) (0.212)Percentage Industrial 0.162 0.204

(0.207) (0.170)Conduit Deal Dummy -0.00325 -0.0566 0.0538 -0.0196

(0.0698) (0.0817) (0.0890) (0.0776)Number of Loans 0.0121∗∗ 0.00758 0.00482 0.00130

(0.00512) (0.00621) (0.00596) (0.00715)Average Loan Size -0.000347 0.000880 -0.000339 0.000671

(0.000599) (0.000655) (0.000610) (0.000725)Geographic Controls no no yes no yesIssuer Fixed Effects yes yes yes yes yesQuarter Fixed Effects no no no yes yes

Observations 696 564 457 564 457

Standard errors in parentheses∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01


Table 4.4: The empirical relationship between Percent AAA and reputation. The smallnumber of observations for models (2) - (5) is due to data availability and will be changedin future versions of the paper. The table displays results of a tobit estimation of (4.1) withvarious controls and the dependent variable Percent AAA. The main coefficient of interest isthat on Total Issuer Downgrades, which is not significant any specification except for model(4). Standard errors in parenthesis are clustered at the Issuer level. All models have χ2

statistics that imply significance at the .001 level.

Percentage AAA

(1) (2) (3) (4) (5)

Constant 0.862∗∗∗ -3.085∗∗∗ -1.266∗∗ 1.191∗∗∗ 1.033∗∗

(0.0000998) (0.794) (0.521) (0.208) (0.475)Days Since Start of Sample 0.286∗∗∗ 0.181∗∗∗

(0.0588) (0.0480)Total Issuer Downgrades 0.0236∗∗∗ -0.00979 0.00265 -0.0143∗∗ 0.00392

(0.00641) (0.00705) (0.00454) (0.00711) (0.00431)Issuer Days in Sample -0.271∗∗∗ -0.169∗∗∗ -0.0711∗∗ -0.00578

(0.0592) (0.0479) (0.0332) (0.0448)Weighted Average Spread -1.018 -1.659∗∗ 0.240 2.252

(1.463) (0.742) (2.676) (1.670)Weighted Average Spread Squared 8.296 -0.933 3.618 -26.28

(20.51) (11.05) (29.64) (16.84)Weighted Average LTV -0.339 -1.068 -0.319 -0.822

(0.359) (1.080) (0.270) (0.968)Weighted Average LTV Squared 0.296 0.830 0.280 0.653

(0.428) (0.878) (0.343) (0.785)Debt Service Coverage Ratio 0.0226 0.0104 0.0243 0.0143

(0.0171) (0.0122) (0.0205) (0.0149)Percentage Office -0.0848 -0.124∗

(0.0599) (0.0675)Percentage Hotel -0.151∗∗ -0.146∗

(0.0742) (0.0835)Percentage Industrial 0.00907 0.0376

(0.126) (0.105)Conduit Deal Dummy 0.161∗∗∗ 0.0758∗ 0.170∗∗∗ 0.0756∗∗

(0.0426) (0.0418) (0.0395) (0.0353)Number of Loans -0.00101 0.00141 -0.000550 0.00138

(0.00258) (0.00304) (0.00199) (0.00291)Average Loan Size -0.0000804 0.000922∗∗ -0.0000193 0.000899∗

(0.000223) (0.000447) (0.000227) (0.000458)Geographic Controls no no yes no yesIssuer Fixed Effects yes yes yes yes yesQuarter Fixed Effects no no no yes yes

Observations 696 564 457 564 457

Standard errors in parentheses∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01


market prices of ABS or evaluate policy measures aimed at increasing the efficiency of thesemarkets should condition on some measure of issuer reputation.

Aside from the results, a principal contribution of this chapter is to introduce two newmeasures: a measure of CMBS issuer reputation, and a measure of CMBS issuer retention.The reputation measure uses performance, rather than market, based information in that Isay that an issuer with more downgrades on past deals has a lower reputation for honesty, allelse equal. The rentention measure looks at capital structure in order to proxy for retentionby assuming that the issuer retains unrated tranches and that the informationally sensitiveportion of the deal is made up of the tranches rated below investment grade. Thus, issuerretention is the percentage of below investment grade principal that is unrated. This measurecan be useful in other empirical work, especially in the understudied area of testing signalingbased thoeries of security design.

Some implications of the model of Chapter 3 were not explored in this chapter and areleft for future work. For instance, the model predicts that prices should be U-shaped in issuerreputation and that issuers will be less honest as their reputations improve, conditional onbeing opportunistic. Unfortunately, price data for CMBS markets is not widely available,and so further proxies must be formed. However, one could also examine the predictive powerof issuer retention on subsequent deal performance to address the second implication.

75

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78

Appendix A

Proofs for Chapter Two

Proof of Proposition 1. Let X∗t denote the proposed contract. First observe that X∗t satisfiesconstraints (PC) since

E

[∫ ∞0

e−γtdX∗t∣∣e = 1

]= E

[e−γt0I(t0 ≤ τ1)y0|e = 1

]= E[e−γt0I(t0 ≤ τ1)e(γ+NλL)t0(a0 + C)|e = 1]

= P (t0 ≤ τ1|e = 1)eNλLt0(a0 + C)

= a0 + C

≥ a0 + C.

Next observe that the proposed contract satisfies constraint (IC) since

E

[∫ ∞0

e−γtdX∗t∣∣e = 0

]= E[e−γt0I(t0 ≤ τ1)y0|e = 0]

= E[e−γt0I(t0 ≤ τ1)e(γ+NλL)t0(a0 + C)|e = 0]

= P (t0 ≤ τ1|e = 0)eNλLt0(a0 + C)

= e−N(λH−λL)t0(a0 + C) = a0.

Now suppose Xt is an arbitrary incentive compatible contract. We will first find aconvenient decomposition of Xt. For n = 0 . . . N let xn(s, t) = XtI(ωn) where ωn ∈ s0 =τ0, . . . , sn = τn, t ≤ τn+1 for n = 1, . . . , N . For convenience let τ0 = 0 and τN+1 =∞. Notethat xn(s, t) : 0 × Rn+ × [sn,∞) → R+ is not a random variable, rather it is a functionof the vector s and time which denotes the payment at time t given the first n defaultsoccurred at times given by the vector s and the (n+ 1)th default has not yet occurred. Thisdecomposition allows us to write

Xt =N∑n=0

xn(τ0, . . . , τn)I(τn < t ≤ τn+1, t). (A.1)

Appendix A. Proofs for Chapter Two 79

First we consider n = 0. We have

E

[∫ τ1

0

e−γtdx0(t)∣∣e = 0

]=

∫ ∞0

e−(γ+NλH)td0x(t)

=

∫ ∞0

eN(λL−λH)te−(γ+NλL)tdx0(t)

≥ e−N(λH−λL)t0

∫ ∞0

(N(λH − λL)(t0 − t) + 1) e−(γ+NλL)tdx0(t)

=a0

a0 + CE

[∫ τ1

0

(N(λH − λL)(t0 − t) + 1) e−γtdx0(t)∣∣e = 1

],

(A.2)

where the second to last step follows from the fact that e−N(λH−λL)t ≥ e−N(λH−λL)t0(N(λH −λL)(t0 − t) + 1) since e−N(λH−λL)t is convex.

Before proceeding, we must find the joint distribution of τ1, . . . , τn and the conditionaldistribution of τn+1 given τ1, . . . , τn given λ = λi for i ∈ L,H. By equation (2.2.2) ofDavid and Nagaraja (2003) the joint probability density function of of τ1, . . . , τn is

fτ1,...,τn(s1, . . . , sn) =λniN !

(N − k)!exp

(−

n∑k=1

λisn

).

By equation (2.2.5) of David and Nagaraja (2003), the increment τn+1− τn is independent ofτ1, . . . , τn and is distributed as a exponential random variable with parameter (N−n)λi. Forn = 1, . . . , N , let An = s ∈ R(n+1)+|0 = s0 ≤ s1 ≤ · · · ≤ sn and dAn = dsn · · · ds1 so thatthe set An denotes the set of possible outcomes of (τ0, . . . , τn). We now calculate the timet = 0 expectation of the present value of the payment attributable to xn(τ0, . . . , τn, t)I(τn <


t ≤ τn+1). We have

E

[∫ τn+1

τn

e−γtdxn(τ0, . . . , τn, t)∣∣e = 0

]= E

[E

[∫ τn+1

τn

e−γtdxn(τ0, . . . , τn, t)∣∣τ1, . . . , τn, e = 0

] ∣∣e = 0

]= E

[(N − n)λhe

γτn

∫ ∞τn

e−(γ+(N−n)λH)(t−τn)dxn(τ0, . . . , τn, t)∣∣e = 0

]=

λn+1H N !

(N − n− 1)!

∫Aneγτn

∫ ∞sn

exp

(−(γ + (N − n)λH)(t− sn)−

n∑k=1

λHsk

)dxn(s, t)dAn

=λn+1H N !

(N − n− 1)!

∫An

∫ ∞sn

exp

(−(γ +NλH)t+ (N − n)λHsn +

n∑k=1

λH(t− sk)

)dxn(s, t)dAn

≥ λn+1L N !

(N − n− 1)!

∫An

∫ ∞sn

exp

(−(γ +NλH)t+ (N − n)λHsn +

n∑k=1

λH(t− sk)

)dxn(s, t)dAn

≥ λn+1L N !

(N − n− 1)!

∫An

∫ ∞sn

exp

(−(γ +NλH)t+ (N − n)λLsn +

n∑k=1

λL(t− sk)

)dxn(s, t)dAn

=λn+1L N !

(N − n− 1)!

∫An

∫ ∞sn

e−N(λH−λL)t exp

(−(γ +NλL)t+ (N − n)λLsn +

n∑k=1

λL(t− sk)

)dxn(s, t)dAn

≥ e−N(λH−λL)t0×

λn+1L N !

(N − n− 1)!

∫An

∫ ∞sn

(N(λH − λL)(t0 − t) + 1) exp

(−(γ +NλL)t+ (N − n)λLsn +

n∑k=1

λL(t− sk)

)dxn(s, t)dAn

=a0

a0 + CE

[∫ τn+1

τn

(N(λH − λL)(t0 − t) + 1)e−γtdxn(τ0, . . . , τn, t)∣∣e = 1

]. (A.3)

The second to last step follows from the fact that e−N(λH−λL)t is convex. The last step followsfrom noting that the last term in the integrand is exactly the probability density functionused in the first step replacing λH with λL and using the definition of t0.

Note that by construction we have

dXt =N∑n=0

dxn(τ0, . . . , τn, t)I(τn < t ≤ τn+1),


hence inequalities (A.2) and (A.3) imply

E

[∫ ∞0

e−γtdXt|e = 0

]=

N∑n=0

E

[∫ τn+1

τn

e−γtdxn(τ0, . . . , τn, t)|e = 0

]

≥N∑n=0

a0

a0 + CE

[∫ τn+1

τn

(N(λH − λL)(t0 − t) + 1)e−γtdxn(τ0, . . . , τn, t)∣∣e = 1

]≥ a0

a0 + CE

[∫ ∞0

(N(λH − λL)(t0 − t) + 1)e−γtdXt|e = 1

]. (A.4)

Combining constraint (IC) with inequality (A.4) we have(1− a0

a0 + C

)E

[∫ ∞0

e−γtdXt|e = 1

]−C ≥ a0

a0 + CE

[∫ ∞0

N(λH − λL)(t0 − t)e−γtdXt|e = 1

].

(A.5)For convenience let E

[∫∞0e−γtdXt|e = 1

]= a0 +C. Rearranging (A.5) and canceling terms

when possible, we are left with the following inequality

E

[∫ ∞0

(t− t0)e−γtdXt|e = 1

]≥ − C(a0 − a0)

a0N(λH − λL). (A.6)

Now consider the cost of this contract to the investors. Using a similar argument asabove and the fact that e−(γ−r)t is convex we have

E

[∫ ∞0

e−rtdXt|e = 1

]≥ e(γ−r)t0

[E

[∫ ∞0

(γ − r)(t− t0)e−γtdXt|e = 1

]+ E

[∫ ∞0

e−γtdXt|e = 1

]](A.7)

≥ e(γ−r)t0[−C(a0 − a0)(γ − r)

a0N(λH − λL)+ a0 + C

](A.8)

= e(γ−r)t0[(

1− C(γ − r)a0N(λH − λL)

)(a0 − a0) + a0 + C

]. (A.9)

Now note that if a0 ≥ C(γ−r)N(λH−λL)

then a0 = a0 by definition and thus (PC) requires

a0 ≥ a0. Alternatively, if a0 <C(γ−r)

N(λH−λL), then a0 = C(γ−r)

N(λH−λL)by definition. In either case we

have

E

[∫ ∞0

e−rtdXt|e = 1

]≥ e(γ−r)t0(a0 + C). (A.10)

But e−(γ−r)t0(a0 +C) is the cost to the investors of the proposed contract. So we have shownthat the proposed contract costs less (or the same) to the investors than any alternativecontract that satisfies (IC) and (PC), hence the proposed contract is optimal.


Proof of Proposition 2. Before we verify equation (2.19) we prove the following combinatorialidentity.

Lemma 2. For all a, b > 0 real numbers and n ≥ 0 an integer

n∑m=0

(n

m

)(−1)n−m

a−mb=

n!bn∏nm=0(a−mb)

. (A.11)

Proof. We proceed by induction on n. For n = 0 the identity is trivial. Now assume theidentity holds for n− 1 so that

n−1∑m=0

(n− 1

m

)(−1)n−1−m

a−mb=

(n− 1)!bn−1∏n−1m=0(a−mb)

. (A.12)

We have

n∑m=0

(n

m

)(−1)n−m

a−mb=

(−1)n

a+

1

a− nb+

n−1∑m=1

(n

m

)(−1)n−m

a−mb

=(−1)n

a+

1

a− nb+

n−1∑m=1

((n− 1

m

)+

(n− 1

m− 1

))(−1)n−m

a−mbby Pascal’s Rule

=n∑

m=1

(n− 1

m− 1

)(−1)n−m

a−mb−

n−1∑m=0

(n− 1

m

)(−1)n−1−m

a−mb

=n−1∑m=0

(n− 1

m

)(−1)n−1−m

(a− b)−mb−

n−1∑m=0

(n− 1

m

)(−1)n−1−m

a−mb

=(n− 1)!bn−1∏n−1

m=0((a− b)−mb)− (n− 1)!bn−1∏n−1

m=0(a−mb)by indentity for n− 1

=(n− 1)!bn−1∏nm=1(a−mb)

− (n− 1)!bn−1∏n−1m=0(a−mb)

=(n− 1)!bn−1(a− (a− nb))∏n

m=0(a−mb)

=n!bn∏n

m=0(a−mb).


We now verify equation (2.19).1 We have

F (δ,Λ, n) = E

[n∑k=1

∫ τk

0

ue−δtdt|e = i

]

=u

δE

[n∑k=1

(1− e−δτk)|e = i

]

=u

δ

n∑k=1

N !

(k − 1)!(N − k)!

∫ ∞0

Λ(1− e−Λt)k−1e−(N−k)Λte−Λt(1− e−δt)dt

=u

δ

n∑k=1

(1− N !

(k − 1)!(N − k)!

∫ ∞0

Λ(1− e−Λt)k−1e−(δ+(N−k+1)Λ)tdt

)

=u

δ

n∑k=1

(1− N !

(k − 1)!(N − k)!

k−1∑m=0

(k − 1

m

)∫ ∞0

Λe−(δ+(N−m)Λ)tdt

)

=u

δ

n∑k=1

(1− N !

(k − 1)!(N − k)!

k−1∑m=0

(k − 1

m

)Λ(−1)k−1−m

δ + (N −m)Λ

)

=u

δ

(n−

n∑k=1

N !Λk

(N − k)!

1∏k−1m=0(δ + (N −m)Λ)

).

Finally we show that the contract given by equations (2.20) is optimal among all in-centive compatible first loss piece contracts. Let the price of the senior tranche be denotedS(n). Over this restricted contract space, the investors problem becomes

minnS(n) + F (r, λL, n)

s.t. S(n) + F (γ, λL, n) ≥ a0 + C

S(n) + F (γ, λH , n) ≥ a0,

1We will use the probability density function of τk for k = 1, . . . , N given λ = Λ. τk is just the kth orderstatistic of a sample of N independent identically distributed exponential random variables with parameterΛ ∈ λL, λH. Let F and f denote the cumulative distribution function and probability density functionrespectively of an exponential random variable with parameter Λ. By equation (2.1.2) of David and Nagaraja(2003) the probability density function of τk is

fτk(t) =N !

(k − 1)!(N − k)!F (t)k−1(1− F (t))N−kf(t)

=N !λ

(k − 1)!(N − k)!Λ(1− e−Λt)k−1e−(N−k)Λte−Λt.


which simplifies to

minnF (r, λL, n) (A.13)

s.t. F (γ, λL, n)− F (γ, λH , n) ≥ C.

Since F (r, λL, n) is increasing in n, the solution of problem (A.13) is given by the smallestn such that F (γ, λL, n)− F (γ, λH , n) ≥ C, which is the desired result.

Proof of Proposition 3. Let Xt be defined as

dXt = dXt − I(t = 0)K.

We can rewrite the optimal contracting problem as

b(a0) = mindXt≥0

E

[∫ ∞0

e−rtdXt

∣∣e = 1

],

such that

E

[∫ ∞0

e−γtdXt

∣∣e = 0

]≤ E

[∫ ∞0

e−γtdXt

∣∣e = 1

]− C,

a0 ≤ E

[∫ ∞0

e−γtdXt

∣∣e = 1

]− C.

The above problem is identical to Definition 1, thus the solution follows directly from Propo-sition 1.

Proof of Proposition 4. Suppose Xt is the contract detailed in Proposition 1. We have

E

[∫ ∞0

e−γtdXt|e = m

]− c ·m = P (τ1 ≥ t0|e = m)e−γt0y0 − c ·m

= e−(mλL+(N−m)λH)t0e(NλL)t0(a0 + c ·N)− c ·m= e−(N−m)(λH−λL)t0(a0 + c ·N)− c ·m

=

(a0 + c · n

a0

)−N−mN

(a0 + c · n)− c ·m

= a0

((a0 + c ·N

a0

)mN

− c ·ma0

)

< a0

(m

N

(a0 + c ·N

a0

− 1

)+ 1− c ·m

a0

)= a0

= E

[∫ ∞0

e−γtdXt|e = 0

].


So that e = 0 is the most profitable deviation from e = N given Xt. This implies thatonly the original incentive compatibility constraint binds. Hence, the result follows fromProposition 1

86

Appendix B

Proofs for Chapter Three

Proof of Proposition ??. Note that since φt = 0 for all t ≥ 0 by Bayes rule, hence it isenough to consider the strategies of the opportunistic type issuer. First suppose there existsa truth telling equilibrium given by Qt(Ht) and πt = 1. Let Q be defined as follows

Q = supQt(Ht)|t ≥ 0 and Ht ∈ Ht. (B.1)

Note that Q > q. For all ε > 0, there exist (t,Ht) such that Q−Qt(Ht) < ε by the definitionof supremum. Let (t,Ht) be such a pair. The one-shot deviation principal implies that

Qt(Ht) + (1−Qt(Ht))γ`− ` ≤ Lt (B.2)

where Lt is the discounted loss faced by the issuer after misreporting a bad type asset giventhe history Ht. By the definition of Q, I have

Lt ≤ γ

(λ(Q+ (1− Q))γ + (1− λ)`

1− γ− λ(q + (1− qγ) + (1− λ)`

1− γ

)= γλ(Q− q). (B.3)

ButQt(Ht) + (1−Qt(Ht))γ`− ` = (1− γ`)(Q− ε− q). (B.4)

This implies that for all ε > 0

(1− γ`)(Q− ε− q) ≤ γλ(Q− q) (B.5)

which implies γ ≥ 1λ+`

since Q > q.

Now suppose γ ≥ 1λ+`

. I’ll show that a truth telling equilibrium is given by Qt = 1,πt = 1, P (q,Ht) = 1 if no misreports have been made and

P (q,Ht) =

1 if q ≤ q` if q > q

Appendix B. Proofs for Chapter Three 87

otherwise. Note that the continuation value of the issuer just depends on whether or notshe has made a misreport. Let L be the discounted loss in continuation value faced by theissuer if she misreports a bad type asset then

L = γ

(λ+ (1− λ)`

1− γ− λ(q + (1− q)γ + (1− λ)`

1− γ

)= γλ(1− q). (B.6)

To see that the above strategies indeed constitute an equilibrium observe that

L = γλ(1− q) = γλ1− `

1− γ`≥ 1− `.

Thus the discounted loss in continuation value is greater than or equal to the one-shotgains from misreporting a bad type asset and the above strategies constitute a truth tellingequilibrium for φ0 = 0.

Proof of Proposition 6.

Separating Equilibrium: Let q ≤ `(1−γ)1−γ` and

P (q, 0) =

1 for q ≤ q` for q ≤ q

,

then

qP (q, 0) + (1− q)γ` = γ`+ (1− γ`)q ≤ `

qP (q, 0) + (1− q)γ ≤ qP (q, 0) + (1− q)γqP (q, 0) + (1− q)γ > γ.

for all q ≤ q. Thus q = arg maxqqP (q, 0) + (1− q)γ and

(1, 1) ∈ arg maxq,ππ`+ (1− π)qP (q, 0) + (1− q)γ`

Thus the issuer chooses to issue q when she has a good type asset and 1 when shehas a bad type asset. Thus conditions (1) and (2) of Definition 4 are satisfied. Thisimplies that E[X|Q, g] = 1 and condition (4) is satisfied, and the strategies proposedin Proposition 6 constitute a separating equilibrium.

Pooling Equilibrium Let γ ≤ λ + (1 − λ)` and P (q, 0) = λ + (1 − λ)` for all q. Then1 = arg maxq qP (q, 0) + (1− q)γ and

(1, 0) ∈ arg maxq,ππ`+ (1− π)qP (q, 0) + (1− q)γ`.


Thus the issuer always chooses to issue a quantity 1 and report that the asset is thegood type. This implies that E[X|Q, g] = λ and condition (4) is satisfied, and thestrategies proposed in Proposition 6 constitute a pooling equilibrium.

Now let γ > λ + (1− λ)` and let Q be the quantity chosen by the issuer in a poolingequilibrium. Then P (q, 0) = λ + (1 − λ)`γ < γ = 0 · P (q, 0) + (1 − 0)γ and q cannotbe an equilibrium quantity strategy for the issuer when she has a good type asset, acontradiction. Thus a pooling equilibrium does not exist.

Proof of Proposition 7. First note that Bayes rule implies that φ = 0 is an absorbing state,since

P(Issuer is honest|Ht) =P(Issuer is honest

⋂Ht)

P(Ht)= 0

so that φt = 0 implies that φs = 0 for all s ≥ t.Now suppose (Q, π) and P (q, 0) is an equilibrium of the static game and not an equi-

librium of the repeated game at time t with φt = 0. Let V be the continuation value theopportunistic issuer receives for playing the strategy (Q, π) given the demand curve P (q, 0).Then at least one of the following must be true:

• Q 6∈ arg maxqqP (q, 0) + (1− q)γ + V

• (Q, π) 6∈ arg maxq,ππ`+ (1− π)(qP (q, 0) + (1− q)γ`) + V

• P (Q, 0) 6= E[Xt+1|Q, g],

any of which implies that (Q, π) and P (q, 0) is not an equilibrium of the static game, acontradiction.

Proof of Lemma 1. For convenience let W (φ) and V (φ) denote the value the honest andopportunistic issuers place on reputation φ respectively. Note that V (0) = W (0) since φ = 0is an absorbing state and both issuer types play the separating equilibrium at φ = 0. Supposethere exists φ and equilibrium strategies QH and QO such that QH(φ) 6= QO(φ), then theinvestors would believe the issuer is the honest type upon observing a quantity equal toQH(φ), and would believe she is the opportunistic type upon observing a level of retentionequal to QO(φ) when the issuer has a prior reputation φ. This implies that P (QH(φ), φ) = 1.The definition of equilibrium strategies for the honest issuer thus implies that

QH(φ) + γ(1−QH((φ)) + γW (1) ≥ QO(φ)P (QO(φ), φ) + γ(1−QO)((φ)) + γW (0)

It is straightforward to shown that π(1) = 0, which implies that V (1) > W (1) so that

QH(φ) + γ(1−QH((φ)) + γV (1) > QO(φ)P (QO(φ), φ) + γ(1−QO)((φ)) + γV (0).


Hence, the opportunistic issuer can profitably deviate to QH(φ) and the strategies QH andQO cannot be played in equilibrium.

Proof of Proposition 8. First suppose there exists a truth telling equilibrium given by Q(φ)and π(φ) = 1, such that Q(φ) > q for some φ. Note that π(φ) = 1 implies φB = φS = φ, sothe one-shot deviation principal then states that for all φ such that Q(φ) > q

Q(φ) + (1−Q(φ))γ`− ` ≤ γ(V (φ)− V (0)). (B.7)

Next note that

Q(φ) + (1−Q(φ))γ`− ` = q + (1− q)γ`− `+ (1− γ`)(Q(φ)− q)= (1− γ`)(Q(φ)− q)

since q + (1− q)γ` = `. Finally, I have

γ(V (φ)− V (0)) = γλ(1− pB)(Q(φ)− q).

So it must be the case that 1− γ` ≤ γλ since Q(φ) > q, which is what I needed to show.Now suppose that γ ≥ 1

λ+`. I show that π(φ) = Q(φ) = 1 for all φ > 0 defines an

equilibrium. To check that these strategies constitute an equilibrium, note that they imply

V (φ) =λ+ (1− λ)`

1− γ(B.8)

for all φ > 0 and φB = φS = φ. This in turn implies

γV (φ)− V (0)) = γλ(1− q)(1− γ)

1− γ

= γλ1− `

1− γ`≥ (1− `),

so that the issuer does not have a profitable one-shot deviation, and the equilibrium isverified.

Proof of Proposition 9. The case for φ0 = 0 is given in Proposition 7. Let V (φ) denote thevalue the issuer receives by playing the strategy (q, 1) forever. Note V (φ) = V for all φwhere V is defined in the proof of Proposition 7, thus the proof applies to φ0 > 0 as well.


Proof of Proposition 10. First I provide the solution to the method of construction given inthe text. Let

φ =1

1− λ

(1− λ(1− γλ)

γ(1− γλ(1− q))

)(B.9)

φ = φB(φB − 1− γ

1− λ

)(B.10)

where

φB =(1− `)− λ(1− γ`)

(1− λ)(1− `). (B.11)

For the remainder of the proof, I assume

1− γ`1− `

<1− γλ

γ(1− γλ(1− q))

so that φB > φ. This assumption could be relaxed with only minor changes to the proof,but doing so would make the calculation of L(φ) for φ ∈ [φ, φ) needlessly laborious. It isstraightforward to show that φ solves equations (3.10)-(3.12) when q∗(φ) = 1, π∗(φ) = 0,and (3.11) binds. Similarly, it is straightforward to show that φ solves (3.10)-(3.12) whenq∗(φ) = 1 and (3.10) binds.

For φ ≥ φ, π∗(φ) = 0 and q∗(φ) = 1, so

p∗(φ) = `+λ(1− `)

1− (1− λ)φ

L1(φ) =γλ(1− `)

1− γλ

(1

1− (1− λ)φ− γ(1− `)

1− γ`

).

For φ ∈ [φ, φ), q∗(φ) = 1 and the solution to (3.16)

π∗(φ) = 1− λ

1− λ

((1− `)

L1(ζ−1(φ))− 1

)1

1− φ, (B.12)

where

ζ(φ) =1

1− λ

(1− λ(1− `)

L1(φ)

)φ. (B.13)

Note that ζ has a unique inverse for φ ≤ φ ≤ φ. Using (B.12), I can calculate p∗(φ) and thefunction L2 for φ ≤ φ ≤ φ:

p∗(φ) = `+ L1(ζ−1(φ)) (B.14)

L2(φ) =γ

1− γλ

(L1(ζ−1(φ))− γ(1− `)2

1− γ`

). (B.15)


Next I consider φ ≤ φ < φ. Let ψn(φ) =∏n

k=0 δk(φ) where δ0(φ) = φ, and

δn(φ) = 1−(

λ

1− λ

)((1− γ)βn−1L(φ)

γ(1− `)2q + (γ − `)βn−1L(φ)

)(B.16)

for n ≥ 0. Then the solution to equations (3.21)-(3.23) is

q∗(φ) = q +βn−1

γ(1− `)L(ψ−1

n (φ)), (B.17)

π∗(φ) = 1−(

λ

1− λ

)((1− γ)βn−1L(ψ−1

n (φ))

γ(1− `)2q + (γ − `)βn−1L(ψ−1n (φ))

)1

1− φ, (B.18)

p∗(φ) = 1− (1− γ)βn−1L(ψ−1n (φ))

γ(1− `)q + βn−1L(ψ−1n (φ))

, (B.19)

L(φ) = βnL(ψ−1n (φ)) (B.20)

where φ(k) = ψk(φ) and n is such that φ(n) ≤ φ < φ(n− 1).To show that the proposed strategies constitute an equilibrium, I must verify that

conditions (1)-(4) of Definition 4 hold. First note condition (4), that investors earn zeroexpected profits in equilibrium, follows by construction. To show that conditions (1)-(3)hold, I repeatedly apply the“one-shot deviation principle.” To see that the principle appliesin this case, note that the game has perfect monitoring so that Proposition 2.2.1 of Mailathand Samuelson (2006) applies. Hence, to show that no profitable deviation exists for theopportunistic issuer, it is enough to examine her single deviation payoffs.

Observe that by construction the function V is the value delivered to the opportunisticissuer by playing the strategies given. Also note that q(1−γ)+γ > ` and ` = q(1−γ`)+γ`,thus to show that the opportunistic issuer never has a profitable one shot deviation it issufficient to show that

q∗(φ)(p∗(φ)− γ) ≥ q(1− γ) (3.10)

q∗(φ)(p∗(φ)− γ`) ≥ (1− γ)`+ γ(V (φB)− V (0)), (3.11)

where

φB =φ

φ+ φ(1− φ).

Also by Lemma 1, I need only consider deviations for the opportunistic issuer, since thehonest issuer will always choose the same quantity. Finally, I must show that the proposedequilibrium prices and strategies yield zero expected profits to the investors. The fact thatequilibrium beliefs are given by Bayes’ rule whenever possible is by construction.

The proof proceeds in three steps by first showing that no profitable one shot deviationexists for the opportunistic issuer over the three subintervals of reputation [φ, 1], [φ, φ), and[0, φ).


Step 1: φ ≥ φ First note that ζ(φ) is increasing for φ ≥ φ since L(φ) is increasing in φ.This implies that p∗(φ) is increasing for φ ≤ φ ≤ φ. More over p∗(φ) is increasing forφ ≥ φ since for such φ

p∗(φ) = `+λ(1− `)

1− (1− λ)φ.

Note that q∗(φ) = 1 so that

p∗(φ)− γ ≥ p∗(φ)− γ

= `+λ(1− `)

λ+ (1− λ)(1− π∗(φ))(1− φ)− γ

= `+ L(ζ−1(φ))− γ = q(1− γ)

by the definition φ. Thus, the opportunistic issuer does not have a profitable one shotdeviation when faced with a good asset. Next note that φB = 1 for all φ ≥ φ

p∗(φ)− γ` ≥ p∗(φ)− γ`

= `(1− γ) +λ(1− `)

1− (1− λ)φ

= (1− γ)`+ γ(V (1)− V (0))

by the definition of φ. For φ ≤ φ < φ

p∗(φ)− γ` = `(1− γ) +λ(1− `)

λ+ (1− λ)(1− π∗(φ))(1− φ)

= `(1− γ) + L(ζ−1(φ))

= (1− γ)`+ L(φB)

by the definition of ζ(φ). Thus, the opportunistic issuer does not have a profitabledeviation when faced with a bad asset.

Step 2: φ ≤ φ < φ

By construction I have

p∗(φ) = γ +q(1− γ)

q∗(φ)

so that q∗(φ)(p∗(φ) − γ) = q(1 − γ), which implies that the opportunistic issuer doesnot have a profitable deviation when faced with a good type asset.

Now consider the case when the opportunistic issuer has a bad type asset. I’ll show thatq∗(φ)(p∗(φ)− γ`) = (1− γ)`+ γ(V (φB)− V (0)) for all φ ≤ φ < φ. Let In = (φn, φn−1]


where φn is the sequence defined by equations (3.17) and (3.18). The argument pro-ceeds by induction on n. For I1, I have

q∗(φ)(p∗(φ)− γ`) = q(1− γ) + γ(1− `)q∗(φ)

= q(1− γ`) + L(ψ−11 (φ))

= (1− γ)`+ γ(V (ψ−11 (φ))− V (0))

and

ψ−11 (φ) =

φ

δ1(ψ−11 (φ))

=φ

φ+ π(φ)(1− φ)= φB

so that q∗(φ)(p∗(φ) − γ`) = (1 − γ)` + γ(V (φB) − V (0)) for all φ ∈ I1. Now assumethat the equality holds for all φ ∈ Ik and all k ≤ n− 1.1 This assumption implies

L(φ) =γ

1− γλ(λ(p∗(φ)q∗(φ) + γ(1− q∗(φ)))

= +(1− λ)(p∗(φ)q∗(φ) + γ(1− q∗(φ))`)− (1− γ)V (0))

=γ

1− γλ(λ(q + γ(1− q)) + (1− λ)(`+ L(φB))− (1− γ)V (0))

=γ(1− λ)

1− γλL(φB)) = βL(φB)

for all φ ∈ Ik and k ≤ n1. Thus L(φ) = βn−1L(ψ−1n−1(φ)) for all φ ∈ In−1. For φ ∈ In,

note that

φB =φ

φ+ π(φ)(1− φ)

=φ

δn(ψ−1n (φ))

= ψn−1(ψ−1n (φ))

by the definition of π, δn and ψn, so that φB ∈ In−1 and L(φB) = βn−1L(ψ−1n−1(φB)).

But ψ−1n−1(φB) = ψ−1

n−1(ψn−1(ψ−1n (φ))) = ψ−1

n (φ) so that

(1− γ)`+ L(φB) = (1− γ)`+ βn−1L(ψ−1n (φ))

= q∗(φ)(p∗(φ)− γ`).

by the definition of q∗ and p∗. Then, by induction, q∗(φ)(p∗(φ)−γ`) = (1−γ)`+L(φB))for all φ ∈ In for all n ≥ 1, and the opportunistic issuer does not have a profitabledeviation when faced with a bad type asset for φ ≤ φ ≤ φ.

1This proof utilizes complete induction so the base case is unnecessary, but is left for clarity.


Step 3: φ < φ The fact that the opportunistic issuer does not have a profitable one shotdeviation for φ < φ follows directly from Proposition 9.

Proof of Proposition 12. First I construct a candidate equilibrium. As in Section 3.3.4, Iassume there exists φ′ and φ′ so that (3.26) binds for φ′ ≤ φ ≤ φ′ and π∗(φ) = 0 for φ ≥ φ′.

Thus for φ ≥ φ′ it is trivial to compute the equilibrium loss and price functions

p∗(φ) = `+λ(1− `)

1− (1− λ)φ(B.21)

L(φ) =γ

1− γλ

(λ(1− `)

1− (1− λ)φ− λ(γ − `)

)(B.22)

The next task is to calculate π∗, and subsequently p∗ and L, for φ′ ≤ φ ≤ φ. Letψn(φ) =

∏nk=0 δk(φ) where δ0(φ) = φ and

δn(φ) = 1− λ

1− λ

1− `(γ

1−γλ

)nL(φ)− λ(γ − `)

∑nk=1

(γ

1−γλ

)k − 1

(B.23)

for n ≥ 1. Let a sequence be given by φ(n) = ψn(φ′), note that now the first element of thesequence is φ′ rather than φ, since it is no longer necessary to consider 0 < Q < 1. Then acandidate equilibrium is given by

π∗(φ) = 1− λ

1− λ

1− `(γ

1−γλ

)nL(ψ−1

n (φ))− λ(γ − `)∑n

k=1

(γ

1−γλ

)k − 1

1

1− φ(B.24)

p∗(φ) = `+

(γ

1− γλ

)nL(ψ−1

n (φ))− λ(γ − `)n∑k=1

(γ

1− γλ

)k. (B.25)

L(φ) =

(γ

1− γλ

)nL(ψ−1

n (φ))− λ(γ − `)n∑k=1

(γ

1− γλ

)k(B.26)

where n is such that maxφn, φ′ ≤ φ ≤ φn−1 and

φ′ = minφ|p∗(φ) ≥ γ.

The rest of the proof proceeds almost identically to the proof of Proposition 10

Proof of Proposition 13. First suppose there exists a truth telling equilibrium given by Q(φ)and φ(φ) = 1 such that Q(φ) > q for some φ. Note that π(φ) = 1 implies φB = φS = φ, sothe one-shot deviation principal then states that for all φ > 0 such that Q(φ) > q

Q(φ) + (1−Q(φ))γ`− ` ≤ γ(1− pB)(V (φ)− V (0)). (B.27)


Next note that

Q(φ) + (1−Q(φ))γ`− ` = q + (1− q)γ`− `+ (1− γ`)(Q(φ)− q)= (1− γ`)(Q(φ)− q)

since q + (1− q)γ` = `. Finally, I have

γ(1− pB)(V (φ)− V (0)) = γλ(1− pB)(Q(φ)− q).

So it must be the case that 1 − γ` ≤ (1 − pB)γλ sinceQ(φ) > q, which is what I needed toshow.

Now suppose that γ ≥ 1(1−pB)λ+`

. I show that π(φ) = Q(φ) = 1 for all φ > 0 defines anequilibrium. To check these strategies constitute an equilibrium, note that they imply

V (φ) =λ+ (1− λ)`

1− γ(B.28)

for all φ > 0 and φB = φS = φ. This in turn implies

γ(1− pB)(V (φ)− V (0)) = γλ(1− pB)(1− q)(1− γ)

1− γ

= γλ(1− pB)1− `

1− γ`≥ (1− `),

so that the issuer does not have a profitable one-shot deviation, and the equilibrium isverified.

Proof of Proposition 14. Suppose that there exists a truth telling equilibrium given by Q(φ)and π(φ) = 1 such that Q(φ) > q for some φ. Note that π(φ) = 1 and pG < 1 implyφB = φS = φF = φ, so the one-shot deviation principal then states that for all φ > 0 thatQ(φ) > q

Q(φ) + (1−Q(φ))γ`− ` ≤ γ(1− pB)(V (φ)− V (φ)) = 0. (B.29)

a contradiction since Q(φ) + (1−Q(φ))γ > q + (1− q)γ` = `.

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Moral Hazard, Adverse Selection, and Mortgage Markets